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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 58 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2 Space and queries .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 58 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3 Approximation ratio .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 59 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4 Generalization: sum of integers .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 59 5 Bibliography 61 2 Overview of the course The goal of this course is to introduce algorithmic techniques for dealing with massive data: data so large that it does not fit in the computer’s memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Broadly speaking, there are two main solutions to deal with massive data: (lossless) compressed data structures and (lossy) data sketches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' A compressed data structure supports fast queries on the data and uses a space proportional to the compressed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This solution is typically lossless: the representation allows to fully reconstruct the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Here we exploit the fact that, in some applications, data is extremely redundant and can be considerably reduced in size (even by orders of magnitude) without losing any information by exploiting this redundancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Interestingly it turns out that, usually, computation on compressed data can be performed faster than on the original (uncompressed) data: the field of compressed data structures exploits the natural duality between compression and computation to achieve this goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The main results we will discuss in the course, compressed dictionaries and compressed text indexes, find important applications in information retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' They allow to pre-process a large collection of documents into a compressed data structure that allows locating substrings very quickly, without decompressing the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' These techniques today stand at the core of modern search engines and sequence-mapping algorithms in computational biology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Importantly, lossless techniques cannot break the information-theoretic lower bound for representing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For example, since the number of subsets of cardinality n of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , u} is �u n � , the information-theoretic lower bound for storing such a subset is log2 �u n � = n log(u/n) + O(n) bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' No lossless data structure can use asymptotically less space than this bound for all such subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The second solution is to resort to lossy compression and throw away some features of the data in order to reduce its size, usually breaking the information-theoretic lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In Chapter 2 we show that any set of cardinality n can be stored with a filter data structure in just O(n) bits bits, provided that we accept a small probability that membership queries fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In Chapters 3 and 4 we see how to shrink even more the size of our data while still being able to compute useful information on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The most important concept here is sketching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Using clever randomized algorithms, we show how to reduce large data sets to sub-linear representations: for example, a set of cardinality n can be stored in just O(polylog(n)) bits using a data sketch supporting useful queries such as set similarity and cardinality estimation (approximately and with a bounded probability of error).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Sketches can be computed off-line in order to reduce the dataset’s size and/or speed up the computation of distances (Chapter 3), or on-line on data streams (Chapter 4), where data is thrown away as soon as it arrives (only data sketches are kept).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Material The proofs of these notes have been put together from various sources whose links can be found in the bibliography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The lectures follow also material from Leskovec et al.’s book [21] (Mining of massive data sets), Amit Chakrabarti’s notes on data stream algorithms [5], Gonzalo Navarro’s book [23] (compressed data structures), and Demetrescu and Finocchi’s chapter on algorithms for data streams from the book [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 3 Chapter 1 Basics 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1 Probability theory 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1 Random variables A random variable (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=') X is a variable that takes values from some sample space Ω according to the outcomes of a random phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Ω is also called the support of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Said otherwise, X takes values in Ω according to some probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' A random variable can be discrete if |Ω| is countable (examples: coin tosses or integer numbers), or continuous (for example, if it takes any real value in some interval).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' When considering multiple R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='s with supports Ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , Ωn, the sample space is the Cartesian product of the individual sample spaces: Ω = Ω1 × · · · × Ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In these notes the support of a R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' will either be a set of integers or an interval of real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Distribution functions We indicate with F(x) = P(X ≤ x) the cumulative distribution function of X : the probability that X takes a value in Ω smaller than or equal to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' P(X = x) = f(x) is the probability mass function (for discrete R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='s) or the probability density function (for continuous R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For discrete R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='s, this is the probability that X takes value x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For continuous R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='s, it’s the function satisfying F(x) = � x −∞ f(x) dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Take the example of fair coin tosses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then, X ∈ {0, 1} (0=tail, 1=head) is a discrete random variable with probability mass function P(X = 0) = P(X = 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Events An event is a subset of the sample space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' a set of assignments for all the R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='s under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Each event has a probability to happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For example, A = {0 ≤ X ≤ 1} is the event indicating that X takes a value between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' P(A∪B) is the probability that either A or B happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' P(A∩B) is the probability that both A and B happen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Sometimes we will also use the symbols ∨ and ∧ in place of ∪ and ∩ (with the same meaning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' P(A|B) indicates the probability that A happens, provided that B has already happened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In general, we have: P(A ∩ B) = P(A) · P(B|A) We say that two events A and B are independent if P(A ∩ B) = P(A) · P(B) or, equivalently, that P(A|B) = P(A) and P(B|A) = P(B): the probability that both happen simultaneously is the product of the probabilities that they happen individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Said otherwise, the fact that one of the two events has happened, does not influence the happening of the other event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 4 Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Consider throwing two fair coins, and indicate A = {first coin = head} and B = {second coin = head}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The two events are clearly independent, so P(A ∩ B) = P(A) · P(B) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='5 · 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='25 On the other hand, consider throwing a coin in front of a mirror, and the two events A = {coin = head} and B = {coin in the mirror = head}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We still have P(A) = P(B) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='5 (the events, considered separately, have both probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='5 to happen), but the two events are clearly dependent!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In fact, P(B|A) = 1 ̸= P(B) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' So: P(A ∩ B) = P(A) · P(B|A) = P(A) · 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We can generalize pairwise independence to a sequence of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='s: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3 (k-wise independence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let W = {X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , Xn} be a set of n random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We say that this set is k-wise independent, for k ≤ n, iff P(�k j=1 Xij = xij) = �k j=1 P(Xij = xij) for any subset {Xi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , Xik} ⊆ W of k random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For k = n, we also say that the random variables are fully independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We will often deal with dependent random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' A useful bound that we will use is the following: Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4 (Union bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For any set of (possibly dependent) events {A1, A2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , An} we have that: P(∪n i=1Ai) ≤ n � i=1 P(Ai) The union bound can sometimes give quite uninformative results since the right hand-side sum can exceed 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The bound becomes extremely useful, however, when dealing with rare events: in this case, the probability on the right hand-side could be much smaller than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This will be indeed the case in some of our applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We finally mention the law of total probability: Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='5 (Law of total probability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' If Bi for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , k is a partition of the sample space, then for any event A: P(A) = k � i=1 P(A ∩ Bi) = k � i=1 P(Bi)P(A|Bi) Expected value and variance Intuitively, the expected value (or mean) E[X] of a numeric random variable X is the arithmetic mean of a large number of independent realizations of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Formally, it is defined as E[X] = � x∈Ω x · f(x) for discrete R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='s and E[X] = � +∞ −∞ x · f(x) dx for continuous R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Some useful properties of the expected value that we will use: Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='6 (Linearity of expectation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let ai be constants and Xi be (any) random variables, for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then E[�n i=1 aiXi] = �n i=1 aiE[Xi] Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For simplicity we consider the cases of E[X +Y ] and E[aX].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The claim follows easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' E[X +Y ] is computed using the law of total probability: E[X + Y ] = � i,j(xi + yj)P(X = xi ∧ Y = yj) = � i,j xi · P(X = xi ∧ Y = yj) + � i,j yj · P(X = xi ∧ Y = yj) = � i xi � j P(X = xi ∧ Y = yj) + � j yj � i P(X = xi ∧ Y = yj) = � i xi · P(X = xi) + � j yj · P(Y = yj) = E[X] + E[Y ] and E[aX] = � i a · xi · P(X = xi) = a � i xi · P(X = xi) = a · E[X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 5 Also, the expected value of a constant a is the constant itself: E[a] = a (a constant a can be regarded as a random variable that takes value a with probability 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In general E[X · Y ] ̸= E[X] · E[Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Equality holds if X and Y are independent, though: E[XY ] = � i,j xiyjP(X = xi ∧ Y = yj) = � i,j xiyjP(X = xi)P(Y = yj) = � � i xiP(X = xi) � �� j yjP(Y = yj) � = E[X]E[Y ] More in general (prove it as an exercise): Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' If X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , Xn are fully independent, then E[�n i=1 Xi] = �n i=1 E[Xi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note that in the above lemma pairwise independence is not sufficient: we need full independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The expected value does not behave well with all operations, however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For example, in general E[1/X] ̸= 1/E[X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The expected value of a non-negative R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' can also be expressed as a function of the cumulative distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We prove the following equality in the continuous case (the discrete case is analogous), which will turn out useful later in these notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For a non-negative continuous random variable X, it holds: E[X] = � ∞ 0 P(X ≥ x) dx Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' First, express P(X ≥ x) = � ∞ x f(t) dt: � ∞ 0 P(X ≥ x) dx = � ∞ 0 � ∞ x f(t) dt dx In the latter integral, for a particular value of t the value f(t) is included in the summation for every value of x ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This observation allows us to invert the order of the two integrals as follows: � ∞ 0 � ∞ x f(t) dt dx = � ∞ 0 � t 0 f(t) dx dt To conclude, observe that � t 0 f(t) dx = f(t) · � t 0 1 dx = f(t) · t, so the latter becomes: � ∞ 0 � t 0 f(t) dx dt = � ∞ 0 t · f(t) dt = E[X] The Variance of a R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' X tells us how much the R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' deviates from its mean: V ar[X] = E[(X − E[X])2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The following equality will turn out useful: Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' V ar[X] = E[X2] − E[X]2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' From linearity of expectation: V ar[X] = E[(X − E[X])2] = E[X2 − 2XE[X] + E[X]2] = E[X2] − 2E[X] · E[E[X]] + E[E[X]2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' If Y is a R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=', note that E[Y ] is a constant (or, a random variable taking one value with probability 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The expected value of a constant is the constant itself, thus the above is equal to E[X2] − E[X]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' If X and Y are independent, then one can verify that V ar[X + Y ] = V ar[X] + V ar[Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' More in general, 6 Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' If X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , Xn are pairwise independent, then V ar[�n i=1 Xi] = �n i=1 V ar[Xi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We have V ar[�n i=1 Xi] = E[(�n i=1 Xi)2] − E[�n i=1 Xi]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The first term evaluates to E[( n � i=1 Xi)2] = n � i=1 E[X2 i ] + 2 � i̸=j E[XiXj] Recalling that E[Xi]E[Xj] = E[XiXj] if Xi and Xj are independent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' the second term evaluates to: E[ n � i=1 Xi]2 = � n � i=1 E[Xi] �2 = n � i=1 E[Xi]2 + 2 � i̸=j E[Xi]E[Xj] = n � i=1 E[Xi]2 + 2 � i̸=j E[XiXj] Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' the difference between the two terms is equal to n � i=1 E[X2 i ] − n � i=1 E[Xi]2 = n � i=1 � E[X2 i ] − E[Xi]2� = n � i=1 V ar[Xi] Crucially,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' note that the above proof does not require full independence (just pairwise independence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This will be important later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let X be a R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' and A be an event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The conditional expectation of X conditioned on A is defined as E[X|A] = � x x · P(X = x|A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The law of total probability (Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='5) implies (prove it as an exercise): Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='11 (Law of total expectation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' If Bi for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , k are a partition of the sample space, then for any random variable X: E[X] = k � i=1 P(Bi)E[X|Bi] Bernoullian R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='s Bernoullian R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='s model the event of flipping a (possibly biased) coin: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' A Bernoullian R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' X takes the value 1 with some probability p (parameter of the Bernoullian), and value 0 with probability 1−p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The notation X ∼ Be(p) means that X is Bernoullian with parameter p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' If X ∼ Be(p), then X has expected value E[X] = p and variance V ar[X] = p(1 − p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note that X2 = X since X ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' E[X] = 0 · (1 − p) + 1 · p = p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' V ar[X] = E[X2] − E[X]2 = p − p2 = p(1 − p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note, as a corollary of the previous lemma, that V ar[X] ≤ E[X] and V ar[X] ≤ 1 − E[X] for Bernoullian R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2 Concentration inequalities Concentration inequalities provide bounds on how likely it is that a random variable deviates from some value (typically, its expected value).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' These will be useful in the next sections to calculate the probability of obtaining a good enough approximation with our randomized algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 7 Markov’s inequality Suppose we know the mean E[X] of a nonnegative R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Markov’s inequality can be used to bound the probability that a random variable takes a value larger than some positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' It goes as follows: Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='14 (Markov’s inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For any nonnegative R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' X and any a > 0 we have: P(X ≥ a) ≤ E[X]/a Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We prove the inequality for discrete R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='s (the continuous case is similar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' E[X] = �∞ x=0 x · P(X = x) ≥ �∞ x=a x · P(X = x) ≥ a · �∞ x=a P(X = x) = a · P(X ≥ a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Chebyshev’s inequality Chebyshev’s inequality gives us a stronger bound than Markov’s, provided that we know the random variable’s variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This inequality bounds the probability that the R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' deviates from its mean by some fixed value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='15 (Chebyshev’s inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For any k > 0: P(|X − E[X]| ≥ k) ≤ V ar[X]/k2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We simply apply Markov to to the R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' (X − E[X])2: P(|X − E[X]| ≥ k) = P((X − E[X])2 ≥ k2) ≤ E[(X − E[X])2]/k2 = V ar[X]/k2 Boosting by averaging A trick to get a better bound is to draw s values of the R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' X and average out the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let ˆX = �s i=1 Xi/s (note: this is a R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=') be the average of the s pairwise independent random variables, distributed as X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then: Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='16 (Boosted Chebyshev’s inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For any k > 0 and integer s ≥ 1: P(| ˆX − E[X]| ≥ k) ≤ V ar[X] s · k2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Since the Xi are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='d, we have E[�s i=1 Xi] = s · E[X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For the same reason, V ar[�s i=1 Xi] = s · V ar[X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then: P(| ˆX − E[X]| ≥ k) = P(s| ˆX − E[X]| ≥ s · k) = P(|s ˆX − sE[X]| ≥ s · k) = P(| �s i=1 Xi − E[�s i=1 Xi]| ≥ s · k) ≤ V ar[�s i=1 Xi]/(s · k)2 = s · V ar[X]/(s · k)2 = V ar[X]/(s · k2) 8 Chernoff-Hoeffding’s inequalities Chernoff-Hoeffding’s inequalities are used to bound the probability that the sum Y = �n i=1 Yi of n independent identically distributed (iid) R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='s Yi exceeds by a given value its expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The inequalities come in two flavors: with additive error and with relative (multiplicative) error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We will prove both for completeness, but only use the former in these notes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The inequalities give a much stronger bound w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Markov precisely because we know a particular property of the R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Y (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' it is a sum of iid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.’s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Here we study the simplified case of Bernoullian R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='17 (Chernoff-Hoeffding bound, additive form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , Yn be fully independent Be(p) random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Denote Y = �n i=1 Yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then, for all t ≥ 0: P(Y ≥ E[Y ] + t) ≤ e −t2 2n [one sided, right] P(Y ≤ E[Y ] − t) ≤ e −t2 2n [one sided, left] P(|Y − E[Y ]| ≥ t) ≤ 2e −t2 2n [double sided] Equivalently, let ˆY = Y/n (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' the average of all Yi) be an estimator for p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then, for all 0 < ϵ < 1: P(| ˆY − p| ≥ ϵ) ≤ 2e−ϵ2n/2 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let Xi = Yi − E[Yi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Each Xi is distributed in the interval [−1, 1], has mean E[Xi] = E[Yi − E[Yi]] = E[Yi]−E[Yi] = 0, and takes value 1−E[Yi] = 1−p with probability p and value 0−E[Yi] = −p with probability 1 − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let X = �n i=1 Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In particular, X = Y − E[Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We prove P(X ≥ t) ≤ e −t2 2n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The same argument will hold for P(−X ≥ t) ≤ e −t2 2n , so by union bound we will get P(|Y − E[Y ]| ≥ t) = P(|X| ≥ t) ≤ 2e −t2 2n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let s > 0 be some free parameter that we will later fix to optimize our bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The event X ≥ t is equivalent to the event esX ≥ est, so: P(X ≥ t) = P(es �n i=1 Xi ≥ est) We apply Markov to the (non-negative1) R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' es �n i=1 Xi, obtaining P(X ≥ t) ≤ E[es �n i=1 Xi]/est = E � n � i=1 esXi � /est which, since the Xi’s are fully independent and identically distributed (in particular, they have the same expected value), yields the inequality P(X ≥ t) ≤ � E[esX1] �n /est (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1) The goal is now to bound the expected value E[esX1] appearing in the above quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Define A = 1+X1 2 and B = 1−X1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note that: A ≥ 0 and B ≥ 0 A + B = (1+X1)+(1−X1) 2 = 1 A − B = (1+X1)−(1−X1) 2 = X1 1Note: X might be negative, but esX is always positive, so we can indeed apply Markov’s inequality 9 We recall Jensen’s inequality: if f(x) is convex, then for any 0 ≤ a ≤ 1 we have f(ax + (1 − a)y) ≤ af(x) + (1 − a)f(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note that ex is convex so, by the above three observations: esX1 = es(A−B) = esA−sB ≤ Aes + Be−s = 1+X1 2 es + 1−X1 2 e−s = es+e−s 2 + X1 · es−e−s 2 Since E[X1] = 0, we obtain: E[esX1] ≤ E � es+e−s 2 + X1 · es−e−s 2 � = es+e−s 2 + es−e−s 2 E [X1] = (es + e−s)/2 The Taylor expansion of es is es = 1+s+ s2 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' + s3 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , while that of e−s is e−s = 1−s+ s2 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' − s3 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' odd terms appear with negative sign).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let even and odd denote the sum of even and odd terms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Replacing the two Taylor series in the quantity (es + e−s)/2, we obtain (es + e−s)/2 = (even + odd)/2 + (even − odd)/2 = even = � i=0,2,4,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' si i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' = �∞ i=0 s2i (2i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Now, note that (2i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' = 1 · 2 · 3 · · · i · (i + 1) · · · 2i ≥ i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' · 2i, so E[esX1] ≤ ∞ � i=0 s2i (2i)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' ≤ ∞ � i=0 s2i i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' · 2i = ∞ � i=0 (s2/2)i i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The term �∞ i=0 (s2/2)i i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' is precisely the Taylor expansion of es2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We conclude E[esX1] ≤ es2/2 and Inequality 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1 becomes P(X ≥ t) ≤ � es2/2�n /est = e(ns2−2st)/2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2) Recall that s is a free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In order to obtain the strongest bound, we have to minimize e(ns2−2st)/2 as a function of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This is equivalent to minimizing ns2 − 2st.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The coefficient of the second-order term is n > 0, so the polynomial indeed has a minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In order to find it, we find the root of its derivative: 2ns − 2t = 0, which tells us that the minimum occurs at s = t/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Replacing s = t/n in Inequality 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2, we finally obtain P(X ≥ t) ≤ e−t2/(2n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' If E[Y ] is small, a bound on the relative error is often more useful: Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='18 (Chernoff-Hoeffding bound, multiplicative form).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , Yn be fully independent Be(p) random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Denote Y = �n i=1 Yi and µ = E[Y ] = np.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then, for all 0 < ϵ < 1: 10 P(Y ≥ (1 + ϵ)µ) ≤ e−µϵ2/3 [one sided, right] P(Y ≤ (1 − ϵ)µ) ≤ e−µϵ2/2 [one sided, left] P(|Y − µ| ≥ ϵµ) ≤ 2e−ϵ2µ/3 [double sided] Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We first study P(Y ≥ (1 + ϵ)µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note that µ = np, since our R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='s are distributed as Be(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The first step is to upper-bound the quantity P(Y ≥ t) (later we will fix t = (1 + ϵ)µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let s > 0 be some parameter that we will later fix to optimize our bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The event Y ≥ t is equivalent to the event esY ≥ est, so: P(Y ≥ t) = P(es �n i=1 Yi ≥ est) We apply Markov to the R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' es �n i=1 Yi, obtaining P(Y ≥ t) ≤ E[es �n i=1 Yi]/est = E � n � i=1 esYi � /est which, since the Yi’s are fully independent and identically distributed (in particular, they have the same expected value), yields the inequality P(Y ≥ t) ≤ � E[esY1] �n /est (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3) Replacing t = (1 + ϵ)µ, we obtain P(Y ≥ (1 + ϵ)µ) ≤ � E[esY1] �n /es(1+ϵ)µ = � E[esY1]e−sp(1+ϵ)�n (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4) The expected value E[esY1] can be bounded as follows: E[esY1] = p · es·1 + (1 − p) · es·0 = p · es + 1 − p = 1 + p(es − 1) ≤ ep(es−1) where in the last step we used the inequality 1 + x ≤ ex with x = p(es − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Combining this with Inequality 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4 we obtain: P(Y ≥ (1 + ϵ)µ) ≤ � ep(es−1)e−sp(1+ϵ)�n = � ees−1e−s(1+ϵ)�µ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='5) By taking s = log(1+ϵ) (it can be shown that this choice optimizes the bound), we have ees−1e−s(1+ϵ) = eϵ−log(1+ϵ)(1+ϵ), thus Inequality 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='5 becomes: P(Y ≥ (1 + ϵ)µ) ≤ � eϵ (1 + ϵ)(1+ϵ) �µ = ρ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='6) To conclude, we bound log ρ = µ(ϵ − (1 + ϵ) log(1 + ϵ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We use the inequality log(1 + ϵ) ≥ ϵ 1+ϵ/2, which holds for all ϵ ≥ 0, and obtain: log ρ ≤ µ � ϵ − ϵ(1 + ϵ) 1 + ϵ/2 � = −µϵ2 2 + ϵ ≤ −µϵ2 3 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='7) Where the latter inequality holds since we assume ϵ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Finally, Bounds 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='6 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='7 yield: P(Y ≥ (1 + ϵ)µ) ≤ e−µϵ2/3 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='8) 11 We are left to find a bound for the symmetric tail P(Y ≤ (1 − ϵ)µ) = P(−Y ≥ −(1 − ϵ)µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Following the same procedure used to obtain Inequality 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4 we have P(−Y ≥ −(1 − ϵ)µ) ≤ � E[e−sY1]es(1−ϵ)p�n We can bound the expectation as follows: E[e−sY1] = p · e−s + (1 − p) = 1 + p(e−s − 1) ≤ ep(e−s−1) and obtain: P(−Y ≥ −(1 − ϵ)µ) ≤ � ee−s−1es(1−ϵ)�µ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='9) It can be shown that the bound is minimized for s = − log(1 − ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This yields: P(Y ≤ (1 − ϵ)µ) ≤ � e−ϵ (1 − ϵ)(1−ϵ) �µ = ρ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='10) Then, log ρ = µ(−ϵ − (1 − ϵ) log(1 − ϵ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We plug the bound log(1 − ϵ) ≥ ϵ2/2−ϵ 1−ϵ , which holds for all 0 ≤ ϵ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then, log ρ ≤ µ(−ϵ − (ϵ2/2 − ϵ)) = −µϵ2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This yields P(Y ≤ (1 − ϵ)µ) ≤ e−µϵ2/2 ≤ e−µϵ2/3 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='11) and by union bound we obtain our double-sided bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Equivalently, we can bound the probability that the arithmetic mean of n independent R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='s devi- ates from its expected value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This yields a useful estimator for Bernoullian R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='s (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' the arithmetic mean of n independent observations of a Bernoullian R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note that the bound improves exponen- tially with the number n of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , Yn be fully independent Be(p) random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Consider the esti- mator ˆY = 1 n �n i=1 Yi for the value p (= E[ ˆY ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then, for all 0 < ϵ < 1: P(| ˆY − p| ≥ ϵp) ≤ 2e−ϵ2np/3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2 Hashing A hash function is a function h : U → [0, M) from some universe U (usually, an interval of integers) to an interval of numbers (usually the integers, but we will also work with the reals).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Informally speaking, h is used to randomizes our data and should have the following basic features: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' h(x) should be “as random” as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Ideally, h should map the elements of U completely uniformly (but we will see that this has a big cost).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' h(x) should be quick to compute algorithmically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Ideally, we would like to compute h(x) in time proportional to the time needed to read x (O(1) if x is an integer, or O(n) if x is a string of length n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' h clearly occupies space in memory, since it is implemented with some kind of data structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This space should be as small as possible (ideally, O(1) words of space, or logarithmic space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' h(x) will also be called the fingerprint of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note that, while h accepts as input any value from U, typically the algorithms using h will apply it to much smaller subsets of U (for example, U might be the set of all 232 possible IPv4 addresses, but the algorithm will work on just a small subset of them).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 12 We formalize the notion of hashing as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We define a family H ⊆ [0, M)|U| of functions (each function assigns a value from [0, M) to each of the |U| universe elements) and extract a uniform2 h ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then, we run our algorithm using the chosen h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The expected-case analysis of the algorithm will take into account the structure of H and the fact that h has been chosen uniformly from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' By a simple information-theoretic argument, it is easy to see that in the worst case we need at least log2 |H| bits in order to represent (and store in memory) h: if, for a contradiction, we used t < log2 |H| bits for all h ∈ H, then we would be able to distinguish only among at most 2t < |H| functions, which is not sufficient since (by uniformity of our choice) any h could be chosen from the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Ideally, we would like our hash function to be completely uniform: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1 (Uniform hash function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Assume that h ∈ H ⊆ [0, M)|U| is chosen uniformly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We say that H is uniform if for any x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , x|U| ∈ [0, M), we have P(h = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , x|U|)) = 1 M |U| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' As it turns out, the requirements (1-3) above are in conflict: in fact, it is impossible to obtain all three simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Assume, for example, that our goal is to obtain a uniform hash function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then, for any choice of x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , x|U| ∈ [0, M), P(h = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , x|U|)) = 1 M |U| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' However, this is possible only if H = [0, M)|U|: in any other case, there would exist at least one choice of x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , x|U| ∈ [0, M) such that P(h = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , x|U|)) < 1 M |U| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Therefore, a uniform hash function must take log2 |H| = log2 M |U| = |U| log2 M bits of memory, which is typically too much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' It is easy to devise such a hash function: fill a vector V [1, |U|] with uniform integers from [0, M), and define h(x) = V [x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note that such a hash function satisfies requirements (1) and (2), but not (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In the next subsection we study good compromises that will work for many algorithms: k-uniform (or k-wise independent) and universal hashing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then, we briefly discuss functions mapping U to real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1 k-uniform/universal hashing In this section we work with integer hash functions: h : [1, n] → [0, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' k-uniform (or k-independent or k-wise independent) hashing is a weaker version of uniform hashing: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We say that the family H is k-uniform (or k-independent or k-wise independent) if and only if, for a uniform choice of h ∈ H, we have that P � k� i=1 h(xi) = yi � = M −k for any choice of distinct x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , xk ∈ [1, n] and (not necessarily distinct) y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , yk ∈ [0, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Thus, a uniform H is the particular case where k = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Equivalently: For any choice of distinct x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , xm with m ≥ k, the random variables h(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , h(xm) are k-wise independent (Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' h maps k-tuples uniformly: the k-tuple (h(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , h(xk)) is a uniform random variable over [0, M)k when x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , xk are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In the next sections we will see that k = 2 is already sufficient in many interesting cases: this case is also called two-independent hashing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Another important concept is that of universality: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We say that H is universal if and only if, for a uniform choice of h ∈ H, we have that P (h(x1) = h(x2)) ≤ 1/M for any choice of distinct x1 ̸= x2 ∈ [1, n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 2One (strong) assumption is always needed for this to work: we can draw uniform integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This is actually impossible, since computers are deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' However, there is a vast literature on pseudo-random number generators (PRNG) which behave reasonably well in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We will thus ignore this problem for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 13 Note that this is at most the probability of collision we would expect if the hash function assigned truly random outputs to every key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' It is easy to see that two-independence implies universality (the converse is not true).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Consider the partition of the sample space {h(x2) = y}y∈[0,M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' By the law of total probability (Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='5): P (h(x1) = h(x2)) = � y∈[0,M) P(h(x1) = h(x2) ∧ h(x2) = y) = � y∈[0,M) P(h(x1) = y ∧ h(x2) = y) = � y∈[0,M) M −2 = 1/M Next, we show a construction (not the only possible one) yielding a two-independent hash function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let M ≥ n be a prime number, and define ha,b(x) = (a · x + b) mod M We define our family ˆH as follows: ˆH = {ha,b : a, b ∈ [0, M)} In other words, a uniform ha,b ∈ ˆH is a uniformly-random polynomial of degree 1 over ZM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note that this function is fully specified by a, b, M and thus it can be stored in O(log M) bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Moreover, ha,b can clearly be evaluated in O(1) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In our applications, the primality requirement for M is not restrictive since for any x, a prime number always exists between x and 2x (and, on average, between x and x + ln(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This will be enough since we will only require asymptotic guarantees for M (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' M ∈ Θ(n) is fine).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We now prove 2-uniformity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' two-independence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' ˆH is a two-independent family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Pick any distinct x1, x2 ∈ [1, n] and (not necessarily distinct) y1, y2 ∈ [0, M − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Crucially, note that since x1 ̸= x2 and M ≥ n, then x1 ̸≡M x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' P (h(x1) = y1 ∧ h(x2) = y2) = P(ax1 + b ≡M y1 ∧ ax2 + b ≡M y2) = P � b ≡M y1 − x1 · y2−y1 x2−x1 ∧ a ≡M y2−y1 x2−x1 � (a) = P � b ≡M y1 − x1 · y2−y1 x2−x1 � P � a ≡M y2−y1 x2−x1 � (b) = M −2 Notes: (a) Simply solve the system in the variables a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note that (x2 −x1)−1 exists because x2 ̸≡M x1 and ZM is a field, thus every element (except 0) has a multiplicative inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' (b) a and b are independent random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In general, it can be proved that the family ˆH = ��k−1 i=0 aixi mod M : a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , ak−1 ∈ [0, M) � is k-uniform whenever M ≥ n is a power of a prime number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note that members of this family take O(k log M) bits of space to be stored and can be evaluated in O(k) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Important note for later: a function h ∈ ˆH maps integers from [1, n] to [0, M), with M ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The co-domain size M ≥ n might be too large in some applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' An example is represented by hash 14 tables, see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4: if the domain is the space of all IPv4 addresses, n = 232 and the table’s size must be M ≥ 232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This is too much, considering that typically we will insert d ≪ n objects into the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4 we will describe a technique for reducing the co-domain size of h while still guaranteeing good statistical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2 Collision-free hashing In some applications we will need a collision-free hash function: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' A hash function h : [1, n] → [0, M) is collision-free on a set A ⊆ [1, n] if, for any x1 ̸= x2, x1, x2 ∈ A, we have h(x1) ̸= h(x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In general we will be happy with a function that satisfies this property with high probability: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='6 (with high probability (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=')).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We say that an event holds with high probability with respect to some quantity n, if its probability is at least 1 − n−c for an arbitrarily large constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Equivalently, we say that the event succeeds with inverse-polynomial probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Typically, in the above definition n is the size of the input (for example, we are hashing n objects into an hash table).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' To simplify our analyses, in these notes we will ignore the incredibly small failure probability of events holding with high probability, and simply assume that they happen with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note that this is reasonable in practice: for example, on a small universe with n = 106 (typical universes are much larger than that), the small constant c = 3 already gives a failure probability of 10−18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' It is far more likely that your program fails because a cosmic ray flips a bit in RAM 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We prove: Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' If a family H of functions h : [1, n] → [0, M) is universal and M ≥ nc+2 for an arbitrarily large constant c, then a uniformly-chosen h ∈ H is collision-free on any set A ⊆ [1, n] with high probability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' with probability at least 1 − n−c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Universality means that P(h(x1) = h(x2)) ≤ M −1 for any x1 ̸= x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Since there are at most |A|2 ≤ n2 pairs of distinct elements in |A|, by union bound the probability of having at least one collision is at most n2/M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' By choosing M ≥ nc+2 we have at least one collision with probability at most n−c, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' our function is collision-free with probability at least 1 − n−c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note that, by choosing M ∈ Θ(nc+2), one hash value (as well as the hash function itself) can be stored in log2 M ∈ O(log n) bits and can be evaluated in constant time using the hash family ˆH introduced in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Observe that function ha,b(x) = ax + b mod M is always collision-free with probability 1 on [1, n] whenever M ≥ n and a ̸= 0 (exercise: prove it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3 Hashing integers to the reals Let H be a family of functions h : [1, n] → {x ∈ R | 0 ≤ x ≤ 1} mapping the integers [1, n] to the real interval {x ∈ R | 0 ≤ x ≤ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' To simplify notation, in these notes we will denote the codomain {x ∈ R | 0 ≤ x ≤ 1} with [0, 1] (not to be confused with the interval of integers [1, n] of the domain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The integer/real nature of the set [a, b] will always be clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We say that H is k- uniform iff, for a uniformly-chosen h ∈ H, (h(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , h(xk)) is uniform in [0, 1]k for any choice of distinct x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' It is impossible to algorithmically draw (and store) a uniform function h : [1, n] → [0, 1], since the interval [0, 1] contains infinitely-many numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' However, we can aim at an approximation with any desired degree of precision (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' decimal digits of h(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Here we show how to simulate such a two-independent hash function (enough for the purposes of these notes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 3stackoverflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='com/questions/2580933/cosmic-rays-what-is-the-probability-they-will-affect-a-program 15 We start with a two-independent discrete hash function h′ : [1, n] → [0, M] (just O(log M) bits of space, see the previous subsection) that maps integers from [1, n] to integers from [0, M] and define h(x) = h′(x)/M ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Since h′ is two-independent, also h is two-independent (over our approxima- tion of [0, 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In addition to being two-independent, h′ should be collision-free on the subset of [1, n], of size d ≤ n, on which the algorithm will work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This is required because, on a truly uniform h : [1, n] → [0, 1], we have P(h(x) = h(y)) = 0 whenever x ̸= y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We will be happy with a guarantee that holds with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Recalling that two-independence implies universality, by the discussion of the previous section it is sufficient to choose M ≥ nc+2 to obtain a collision-free hash function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Simplifications For simplicity, in the rest of the notes we will simply say “h is k-wise indepen- dent/uniform, etc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='..” hash function, instead of “h is a function uniformly chosen from a k-wise independent/uniform, etc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' family H”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4 Hash tables Since our hash functions h : [1, n] → [0, M) have good statistical properties (in particular, low collision rate), can we use them as an index in an array H[0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , M −1] to implement a set (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' a dictionary data structure)?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The collision-free property ensures that H[h(x)] is likely to contain only (data associated with) x, so it looks like this is going to be a fast data structure with constant-time insert/access/delete operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The problem we want to solve is: Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='8 (Hash table).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' A hash table over [1, n] is a data structure H implementing a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We want H to support quickly the following operations: Insert x in H Check if x ∈ H Remove x from H After inserting d elements, the space of H should be bounded by O(d) words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We now give an implementation of a hash table: hashing by chaining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Assuming we know the number d of elements that will be inserted in our hash table, we choose a hash function h : [1, n] → [0, d) and initialize an empty vector H[0, d − 1] of size d (indexed from index 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The cell H[i] contains an array—we call it the i-th chain—, initially of size 0 (to be more precise, H[i] is a pointer to a resizable array).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then, operation insert will be implemented by appending x at the end of the chain H[h(x)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Arrays are resized using a doubling technique, so that they occupy linear space and support appending an integer in constant amortized time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note that this implementation allows us to associate with each x ∈ H also some satellite date (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' a pointer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Of course, we want the collision probability of h to be as low as possible: for any x ̸= y, we want P(h(x) = h(y)) ≤ 1/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The function hab : [1, n] → [0, M) of the previous section has M > n, so the space of H would be prohibitively large (n ≫ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We now describe a hash function with codomain size O(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Choose a prime number M > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then, choose two uniform numbers a ∈ (0, M) and b ∈ [0, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Our family of hash functions is: ¯h(x) = ((a · x + b) mod M) mod d Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Function ¯h(x) is universal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' for any x ̸= y, it holds P(¯h(x) = ¯h(y)) ≤ 1/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 16 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let X = (a·x+b) mod M and Y = (a·y+b) mod M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' First note that since M > n and a ̸= 0, then x ̸= y ⇒ X ̸= Y (exercise: prove it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The equality ¯h(x) = ¯h(y) holds when X ≡d Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' It is easy to see that X and Y are uniform random variables with support [0, M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Moreover, with the same reasoning of the proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4, one can see that P(X = i ∧ Y = j) = 1 (M−1)M : X and Y are almost two-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' From this, we get P(Y = j|X = i) = P(X = i ∧ Y = j)/P(X = i) = 1/(M − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Now, fix a given X = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In the interval [0, M), there are at most ⌈M/d⌉ − 1 ≤ (M − 1)/d values for Y such that Y ≡d i: those are all the integers (i excluded) whose distance from i is a multiple of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Since P(Y = j|X = i) = (M − 1)−1, by union bound the chance of picking such a value of Y is at most P(Y ≡d i|X = i) ≤ M−1 d (M − 1)−1 = 1/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We can finally apply the law of total probability on the partition X = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , M−1 of the event space and obtain P(¯h(x) = ¯h(y)) = � i∈[0,M) P(X = i) · P(Y ≡d i|X = i) ≤ � i∈[0,M)(1/M) · 1/d ≤ 1/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , xd be the d elements in the hash table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Universality of ¯h implies E[|H[¯h(xi)]|] = E[� j̸=i 1¯h(xi)=¯h(xj)] = � j̸=i E[1¯h(xi)=¯h(xj)] ≤ d · (1/d) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This is the expected length of xi’s chain (for any i), so each operation on the hash table takes expected O(1) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We can lift the assumption that we know d in advance with a classic doubling technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Initially, we allocate d = 1 cells for H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' After having inserted the d-th element, we allocate a new table of size 2d, re-hash all elements in this new table using a new hash function modulo 2d, and delete the old table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' It is easy to see that the total space is linear and operations still take O(1) amortized time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Expected longest chain We have established that a universal hash function generates chains of expected length O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This means that n insertions in the hash table will take expected O(n) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Another interesting question is: what is the variance of the chain length, and what is the expected length of the longest chain?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This is interesting because this quantity is precisely the expected worst- case time we should expect for one operation (the slowest one) when inserting n elements in a hash of size n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We introduce some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Suppose x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , xd are the elements we want to insert in the hash table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let 1i,j = � � � 1 if h(xi) = j 0 otherwise be the indicator R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' taking value 1 if and only if xi hashes to the j-th hash bucket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The length of the j-th chain is then Lj = �d i=1 1i,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The quantity L′ j = |Lj − E[Lj]| indicates how much Lj differs from its expected value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' since for universal hash functions we have E[Lj] = O(1) (assuming that the hash’ codomain has size d), L′ j = Θ(Lj) so the two R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='s are asymptotically equivalent (we will study L′ j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let L′ max = maxj L′ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Our goal is to study E[L′ max].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' It turns out that, if h is completely uniform, then E[L′ max] ∈ O(log d/ log log d);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' this is the classic balls into bins problem4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Surprisingly, a simple policy (the so-called power of two choices) improves this bound exponentially: let’s use two completely uniform hash functions h1 and h2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We insert each element x either in H[h1(x)] or in H[h2(x)], choosing the bucket that contains the least number of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This simple policy yields E[L′ max] ∈ O(log log d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In practice, however, we almost never use completely uniform hash functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' What happens if h is simply two-independent?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' the following theorem holds for any 2-independent hash function (including ¯h of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='9, even if that function is not completely 2-independent): Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' If h is two-independent, then E[L′ max] ∈ O( √ d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' If h is two-independent, then it is also 1-independent so 1i,j ∼ Be(1/d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then, E[Lj] = d · (1/d) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' From two-independence, we also get V ar[Lj] = V ar[� i 1i,j] = d · V ar[11,j] = d · (1/d) · 4en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='org/wiki/Balls_into_bins_problem 17 (1 − 1/d) = 1 − 1/d ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then, applying Chebyshev: P(L′ j ≥ k) = P(|Lj − E[Lj]| ≥ k) ≤ V ar[Lj]/k2 ≤ 1/k2 By union bound: P(L′ max ≥ k) = P(� j L′ j ≥ k) ≤ d/k2 Let us rewrite k = √ t · √ d: P(L′ max ≥ √ t · √ d) ≤ 1/t We apply the law of total expectation on the partition of the event space [0, √ d), [ √ 2i√ d, √ 2i+1√ d), for all integers i ≥ 0 (assume for simplicity that L′ max ∈ [0, ∞): this does not affect our upper bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The probability that L′ max falls in the interval [ √ 2i√ d, √ 2i+1√ d) is at most 2−i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' moreover, inside this interval the expectation of L′ max is (by definition of the interval) at most √ 2i+1√ d, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' : E[L′ max|L′ max ∈ [ √ 2i√ d, √ 2i+1√ d)] ≤ √ 2i+1√ d Applying the law of total expectation: E[L′ max] ≤ �∞ i=0 2−i√ 2i+1√ d = √ 2d · �∞ i=0 2−i/2 It is easy to derive that �∞ i=0 2−i/2 = 2+ √ 2 (prove it as an exercise), which proves our main claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Alon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' [1] proved that there exist two-independent hash functions with E[L′ max] ∈ Ω( √ d), so the above bound is tight in general for two-independent hash functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Nothing however prevents a particular two-independent hash function to beat the bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In fact, Knudsen in [20] proved that the simple function ¯h of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='9 satisfies E[L′ max] ∈ O( 3√d log d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 18 Chapter 2 Probabilistic filters A filter is a probabilistic data structure encoding a set and supporting typical set operations such as insertion of new elements, membership queries, union/intersection of two sets, frequency estimation (in the case of multi-sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The data structure is probabilistic in the sense that queries such as membership and frequency estimation may return a wrong result with a small (user-defined) probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Typically, the smaller this probability is, the larger the space of the data structure will be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The name filter comes from the typical usage case of these data structures: usually, they are used as an interface to a much larger and slower (but exact) set data structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' the role of the filter is to quickly discard negative queries in order to minimize the number of queries performed on the slower data structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Another usage case is to filter streams: filters guaranteeing no false negatives (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Bloom filters, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1) can be used to quickly discard most stream elements that do not meet some criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' A typical real-case example comes from databases: when implementing a database management system, a good idea could be to keep in RAM a fast (and small) filter guaranteeing no false negatives (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' a Bloom filter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' A membership query first goes through the filter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' the disk is queried if and only if the filter returns a positive answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In situations where the user expects many negative queries, such a strategy speeds up queries by orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Another example is malicious URL detection: for example, the Google Chrome browser uses a local Bloom filter to detect malicious URLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Only the URLs that pass the filter, are checked on Google’s remote servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note: also the sketches discussed in Chapter 3 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' MinHash and CountMinSketch) are a random- ized (approximate) representation of sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' As a matter of fact, CountMinSketch is often introduced as a filter data structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The characterizing difference between those sketches and the filters described in this section, is that the former often require sublinear space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' o(n) bits, where n is the number of elements in the set), while the latter still require linear (O(n) bits) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The common feature of the both solutions is that they break the information-theoretic lower bound of log2 �u n � = n log(u/n)+O(n) bits which are required in the worst case to represent a set of cardinality n over a universe of cardinality u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In general, this is achieved at the price of returning wrong answers with some small probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1 Bloom filters A Bloom filter is a data structure representing a set S under these operations: Insert: given an element x (which may be already in the set S) update the set as S ← S ∪ {x} Membership: given an element x, return YES if x ∈ S and NO otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Bloom filters do not support delete operations (counting Bloom filters do: see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Bloom filters guarantee a bounded one-sided error probability on membership queries, as long as the maximum capacity of the filter is not exceeded: if x ∈ S, a membership query returns YES with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' If 19 x /∈ S, a membership query returns NO with probability 1 − δ, for any parameter 0 < δ < 1 chosen at initialization time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Insert queries always succeed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The filter uses Θ(n log(1/δ)) bits of space to store at most n elements from a universe of cardinality u (n is the filter’s capacity): notice that this space is independent from u and breaks the lower bound of n log(u/n) + O(n) bits when δ is not too small and u is much larger than n (which is typically the case: for example, if the universe is the set of all IPv4 address, then u = 232).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1 The data structure Let h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , hk : U → [0, m) be k hash functions, where U is the universe from which the set elements are chosen (for example: integers, strings, etc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' k and m are two integer parameters that will be chosen later as a function of n and δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We will assume that these functions are independent and completely uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This assumption simplifies the analysis but, as seen in the previous section, it is often not realistic in practice;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' however, when using good hash functions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' cryptographic functions such as SHA-256), the practical performance of Bloom filters are close to those predicted by this idealized setting, so in this particular scenario we will accept the uniformity assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The Bloom filter is simply a bit-vector B[0, m − 1] of length m, initialized with all entries equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Queries are implemented as follows: Insert: to insert x in the set, we set B[hi(x)] ← 1 for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Membership: to check if x belongs to the set, we return �k i=1 B[hi(x)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In other words, the filter returns YES if and only if all bits B[h1(x)], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , B[hk(x)] are equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' It is easy to see that no false negatives can occur: if the Bloom filter returns NO, then the element is not in the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Equivalently, if an element is in the set then the filter returns YES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' However, false positive may occur due to hash collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In the next section we analyze their probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' See florian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='io/bloom-filters/ for a nice online demo of how a Bloom filters works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2 Analysis Notice that the insertion of one element in the set causes the modification k random bits in the bitvector B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Since we assume independence and uniformity for the hash functions, after inserting at most n elements the probability that a particular bit B[i] is equal to 0 is at least � 1 − 1 m �nk = � 1 − 1 m �m· nk m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For m → ∞, this tends to p = e−nk/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let x be an element not in the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The probability that the filter returns a false positive is then at most (1 − p)k = � 1 − e−nk/m�k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' It turns out that this quantity is minimized for k = (m/n) ln 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' replacing this value into the above probability, we get that the false positive probability is at most (1/2)(m/n) ln 2 Solving (1/2)(m/n) ln 2 = δ as a function of m, we finally get m = n log2 e·log2(1/δ) ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='44·n log2(1/δ) and k = (m/n) ln 2 = log2(1/δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' An interesting observation: using k = (m/n) ln 2, after exactly n insertions the probability that any bit B[i] is 0 is (for m → ∞) equal to e−nk/m = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In other words, after n insertions B is a uniform bitvector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This makes sense, because it means that the entropy of B is maximized, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=', we have packed as much information as possible inside it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let 0 < δ < 1 be a user-defined parameter, and let n be a maximum capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' By using k = log2(1/δ) hash functions and m = n log2 e · log2(1/δ) bits of space, the Bloom filters guarantees false positive probability at most δ, provided that no more than n elements are inserted into the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 20 In practice, however, k and m as computed above are often not integer values!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' One solution is to choose k as the closest integer to log2(1/δ) and then choose the smallest integer m such that � 1 − e−nk/m�k ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Suppose we want to build a Bloom filter to store at most n = 107 malicious URLs, with false positive probability δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The average URL length is around 77 bytes (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' supermind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' org/ blog/ 740/ average-length-of-a-url-part-2 ), so just storing these URLs would require around 734 MiB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Choosing k = 3 and m = 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='000, our Bloom filter uses just 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='73 MiB of space (about 5 bits per URL) and returns false positives at most 10% of the times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The filter uses 128 times less space than the plain URLs and speeds up negative queries by one order of magnitude (assuming that the filter resides locally in RAM and the URLs are on a separate server or on a local disk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2 Counting Bloom filters What if we wanted to support deletions from the Bloom filter?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The idea is to replace the bits of the bitvector B with counters of t bits (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' able to store integers in the range [0, 2t)), for some parameter t to be decided later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The resulting structure is called counting Bloom filter and works as follows: Insert: to insert x in the set, we update B[hi(x)] ← B[hi(x)] + 1 for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Delete: to delete x from the set, we update B[hi(x)] ← B[hi(x)] − 1 for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Membership: we return YES if and only if B[hi(x)] ≥ 1 for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' To simplify our analysis, we assume that we never insert an element that is already in the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Similarly, we assume that we only delete elements which are in the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In general, one can check these pre-conditions using the filter and, only if the filter returns a positive answer, the (slow) memory storing the set exactly, so we will assume they hold (note: false negatives will occur with inverse-polynomial probability only, so in practice we can ignore them).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The first observation is that, as long as all m counters are smaller than 2t (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' no overflows occur), the filter behaves exactly as a standard Bloom filter: no false negatives occur, and false positives occur with probability at most δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We therefore choose m = n log2 e·log2(1/δ) and k = (m/n) ln 2 = log2(1/δ) as in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let T = 2t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then, the probability that one particular entry B[i] exceeds value T after n insertions is P (B[i] ≥ T) ≤ �nk T � 1 mT ≤ �enk T �T 1 mT = �enk Tm �T = �e ln 2 T �T ≤ (1/2)T for T ≥ 4 The first inequality above comes from the observation that B[i] ≥ T happens iff at least T of the nk increments (resulting from the n insertions) affect B[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This probability decreases as T increases, so in order to get an upper bound we can focus on the case where exactly T increments affect B[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let’s call these T increments “bad”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Since the T bad increments could be distributed in �nk T � possible ways among the nk increments, and each of these combinations of T bad increments occurs with probability (1/m)T (T independent events of probability 1/m each), by union bound we get the first inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The second inequality comes from the inequality �a b � ≤ � e·a b �b, where e is the base of the natural logarithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The last inequality holds for 2t = T ≥ 4, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' t ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' After n insertions and any number of deletions, the filter could return a false negative on a particular query if at least one of the k counters associated with the query reaches value T at some point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note that we cannot assume these k counters to be independent, since if we know that B[i] ≥ T then at least T of the nk increments have been “wasted” on B[i], thus it is less likely for some other counter B[j], j ̸= i to overflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We therefore use union bound and obtain that a particular query returns a false negative with probability 21 P(false negative) ≤ k(1/2)2t In practice, the value t = 4 (4 bits per counter) is already sufficient to guarantee a negligible probability of false negatives for realistic values of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Suppose we want to build a counting Bloom filter to store at most n = 107 malicious URLs, with false positive probability δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' From Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2, just storing these URLs would require around 734 MiB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Choosing k = 3, m = 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='000, and t = 4, our Bloom filter uses just 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='92 MiB of space (32 times less than the plain URLs) and returns false positives at most 10% of the times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The probability that a query returns a false negative is at most 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='0046%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3 Quotient filters Quotient filters (QF) were introduced in 2011 by Bender et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This filter uses a space slightly larger than classic Bloom filters, with a similar false positive rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In addition, the QF supports deletes without incurring into false negatives and has a much better cache locality (thus being faster than the Bloom filter in practice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In essence, a QF is just a clever (space-efficient) implementation of hashing with chaining and quotienting, see Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We first describe how the filter works by using a standard hash table T[0, m − 1] where each T[i] stores a chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then, in the next subsection we show how to encode T using just one array H of small integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We use a uniform hash function h mapping our universe to [0, 2p), for a value p that will be chosen later 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We break hash values h(x) (of p bits) into two parts: a suffix (remainder) R(x) of r bits (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' the r least significant bits of h(x)) and a prefix (quotient) Q(x) of q = p − r bits (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' the q most significant bits of h(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The chain contains m = 2q entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The value q is chosen such that m = 2q ≥ n (n is the maximum number of elements that will be inserted in the set) and such that the load factor α = n/m of the table, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' the fraction of occupied slots, is a small enough constant (a practical evaluation for different values of α is provided in the paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The operations on this simplified implementation of the filter work as follows: To insert x in the set, we append R(x) to the chain stored in T[Q(x)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Importantly, we allow repetitions of remainders inside the same chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' To remove x from the set, we remove one occurrence of R(x) from the chain stored in T[Q(x)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' To check if x belongs to the set, we check if R(x) appears inside the chain stored in T[Q(x)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Notice that this scheme allows retrieving h(x) from the table: if remainder R is stored in the Q-th chain, then the corresponding fingerprint is Q · 2r + R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In other words, the trick is to exploit the location (Q) inside the hash table to store information implicitly, in order to reduce the information (R) that is explicitly inserted inside the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This trick was introduced by Knuth in his 1973 book “The Art of Computer Programming: Sorting and Searching”, and already allows to save some space with respect to a classic chained hash that stores the full fingerprints h(x) inside its chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Importantly, note that this implementation generates a false positive when we query an element x which is not in the set, and the set contains another element y ̸= x with h(y) = h(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Later we will analyze the false positive probability, which can be reduced by increasing p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note also that, thanks to the fact that we store all occurrences of repeated fingerprints in the table, the data structure does not generate false negatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 1Again, the uniformity assumption is not realistic in practice, but the authors show that, by using “good in practice” hash functions, the practical performance follow those predicted by theory 22 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1: Hashing with chaining and quotienting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' A quotient filter is a space-efficient implementation (avoiding pointers) of this hashing scheme, see Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1 Reducing the space The QF encodes the table T of the previous subsection using a circular2 array H[0, m − 1] of m = 2q slots, each containing an integer of r + 3 bits: r bits storing a remainder, in addition to the following 3 metadata bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' is-occupied[i]: this bit records whether there exists an element x in the set such that i = Q(x), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' if chain number i contains any remainder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' is-shifted[i]: this bit is equal to 0 if and only if the remainder R(x) stored in H[i] corresponds to an element x such that Q(x) = i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' if R(x) belongs to the i-th chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In other words, is-shifted[i]=1 indicates that the remainder R(x) stored in H[i] has been shifted to the right w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' its “natural” position H[Q(x)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' is-continuation[i]: this bit is equal to 1 if and only if the remainder R(x) stored in H[i] belongs to the same chain of the remainder R(y) stored in H[i − 1], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' if the two corresponding set elements x, y are such that Q(x) = Q(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2 shows the QF implementation of the hash table of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In this example, the QF uses in total m · (r + 3) = 40 bits (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' the bitvector to the right of “Metadata + remainders = QF”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' While inserting elements, the following invariant is maintained: if Q(x) < Q(y), then R(x) comes before R(y) in the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We call runs contiguous subsequences corresponding to the same quotient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' See Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2: there are three runs, sorted by their corresponding quotients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We say that a cluster is a maximal contiguous portion H[i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , j] of runs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' in particular, H[i − 1] and H[j + 1] are empty (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' do not store any remainder R(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2, there is just one cluster (the array H is circular, so that after the cluster there is indeed an empty slot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' It is not hard to see that this implementation allows to simulate chaining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Observe that: 2circular means that the cell virtually following H[m − 1] is H[0] 23 000 001 010 011 100 101 110 111 11 01 00 10 10 01Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2: Quotient filter encoding of the hash table in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Empty cells are those such that is occupied[i] = 0 and is shifted[i] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Runs H[i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , i + k] of the same quotient can be identified because is continuation[i] = 0 and is continuation[i + j] = 1 for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Points (1) and (2) allow us identifying clusters and runs inside a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Looking at all ’1’-bits is occupied[i] = 1 inside a cluster, we can moreover reconstruct which quotients Q(x) are stored inside the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note also that R(x) is always stored inside the cluster containing cell H[Q(x)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Since quotients in a cluster are sorted and known, and we know their corresponding runs, it is possible to insert/delete/query an element x by scanning the cluster containing position H[Q(x)] (see the original paper [2] for the detailed algorithms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Point (4) above implies that the average/worst-case query times are asymptotically equal to the average/largest cluster length, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2 Analysis A false positive occurs when we query the QF on an element x not in the set, and h(x) = h(y) for some y in the set (x ̸= y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Since we assume h to be completely uniform, the probability that h(x) = h(y) is 1/2p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then, the probability that h(x) ̸= h(y) is 1 − 1/2p, thus the probability that h(x) ̸= h(y) for all the n elements y in the set is (again by uniformity of h) (1 − 1/2p)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We conclude that the false positive probability is bounded by 1 − � 1 − 1 2p �n = 1 − � 1 − 1 2p �2p· n 2p ≈ 1 − e−n/2p ≤ n 2p ≤ 2q 2p = 2−r where the first inequality (≤) follows from the inequality x ≤ ln � 1 1−x � for x < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' By setting δ = 2−r (where 0 < δ < 1 is the chosen false positive rate), we obtain that the space used by the QF is m · (r + 3) = m · log2(1/δ) + 3m bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Recalling that m = n/α, where 0 < α < 1 is the table’s load factor, we finally obtain that the space is (n/α) · log2(1/δ) + 3n/α bits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The choice of the constant α affects the queries’ running times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In any case, as the following theorem shows, the length of the longest cluster does not exceed Θ(log m) with high probability: 24 000 001 010011 100 101 110 01000 ,6q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' \'random\' "chain" "collision "data" hashing B-occupied is-shiftedis-continuationTheorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For any constant ϵ ≥ 0, the probability that the longest cluster exceeds length 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='5(1 + ϵ) ln m α − ln α − 1 is at most m−ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' If H[i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , i + k − 1] is a cluster, then k elements x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , xk in the set are such that Q(xj) ∈ [i, i + k − 1] for all j = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For a fixed x, the probability that Q(x) ∈ [i, i + k − 1] is k/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Since the hash is fully uniform, the number C of elements that hash inside H[i, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , i + k − 1] is the sum of n independent Bernoulli variables Be(k/m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note that E[C] = n · (k/m) = kα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Applying Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='18 (multiplicative Chernoff), we obtain P(C ≥ k) = P(C ≥ (1 + (1/α − 1))E[C]) ≤ e−kα·(1/α−1)2/3 = e−k·(1/α+α−2)/3 A cluster could start in any of the m cells of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The longest cluster is longer than k iff at least one cluster is longer than k, so by union bound: P(longest cluster length ≥ k) ≤ m · e−k·(1/α+α−2)/3 The above probability is equal to m−ϵ for k = 3(1 + ϵ) ln m 1/α + α − 2 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='5(1 + ϵ) ln m α − ln α − 1 where the last inequality follows from 1/α + α − 2 ≥ 2(α − ln α − 1) for 0 < α ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Moreover, the expected cluster length is a constant: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For constant load factor 0 < α < 1, the expected cluster length is O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' From the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1, the probability P(C = k) that the length of a particular cluster is k is at most e−k·(1/α+α−2)/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Using the fact that �∞ k=1 k · e−kc = ec (ec−1)2 , we obtain that, for constant 0 < α < 1: E[C] ≤ ∞ � k=1 k · e−k·(1/α+α−2)/3 ≤ e(1/α+α−2)/3 (e(1/α+α−2)/3 − 1)2 ∈ O(1) See the original paper [2] for a tighter bound as function of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In practice, choosing α ∈ [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='9] guarantees a good space-time trade-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' By choosing α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='5, for example, the space of the filter is 2nr + 6n = 2n · log2(1/δ) + 6n bits and 99% of the clusters have less than 24 elements (see [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This space is slightly larger than that of the Bloom filter, but query times of the QF are much faster: each query requires scanning only one cluster which (due to the average cluster length) will probably fit into a single cache line, thus causing at most one cache miss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Bloom filters, on the other hand, generate one cache miss per hash function used: this makes them several times slower than Quotient filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 25 Chapter 3 Similarity-preserving sketching Let x be some data: a set, a string, an integer, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' A data sketch is a randomized function f mapping x to a (short) sequence of bits with the following interesting properties: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' f is easy to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The bit-size of f(x) is much smaller than the bit-size of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' f(x) is easy to update if x gets updated (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' we add an element if x is a set, or we append a character if x is a string).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This may include combining sketches (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' to obtain the sketch of the union, if the data represents sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' f(x) can be used to efficiently compute some properties of x (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' the number of distinct elements contained in x, if x is a set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' A similarity-preserving sketch has a somewhat stronger property that allows comparing sketches: if x and y are similar (according to some measure of similarity, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Euclidean distance), then f(x) and f(y) are likely to be similar (according to some measure of similarity, not necessarily the same as before).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note that f(x) and f(y) are (in general, dependent) random variables, being f a randomized function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In general, we will discuss sketches whose sizes are poly-logarithmic with respect to |x| (the size will also depend on other parameters, such as error rate and probability of obtaining a good approximation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1 Sketching for identity - Rabin’s hash function The most straightforward measure of similarity is identity: given x and y, is x equal to y?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Without loss of generality, let x be a string of length n over alphabet Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Without loss of generality, Σ = [0, σ − 1] and we can view strings as integers in base σ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note that this setting can also be used to represent subsets of [1, n], letting Σ = {0, 1} and |x| = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Observe that, for any function f, if |f(x)| < |x| (| · | means “number of bits”) then collisions must occur: there must exist pairs x ̸= y such that f(x) = f(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The first idea to solve the problem could be to use function ¯h(x) = ((a · x + b) mod M) mod d of Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='9: we simply view the string x as a number with |x| digits in base |Σ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Unfortunately, this is not a good idea: recalling that we require M > x for any input x of our function, we would need to perform modular arithmetic on integers with n digits!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Rabin’s hashing is a string hashing scheme that solves the above problem (but it cannot achieve universality — even if it guarantees a very low collision probability, see below): 26 Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1 (Rabin’s hash function 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Fix a prime number q, and pick a uniform z ∈ [0, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let x[1, n] ∈ Σn be a string of length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Rabin’s hash function κq,z(x) is defined as: κq,z(x) = �n−1 � i=0 x[n − i] · zi � mod q In other words: κq,z(x) is a polynomial modulo q evaluated in z (a random point in [0, q)) and having as coefficients the characters of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 2 First, we show that κq,z(x) is easy to compute and update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Suppose we wish to append a character c at the end of x, thereby obtaining the string x·c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' It is easy to see that this can be achieved as follows (Horner’s method for evaluating polynomials): Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' κq,z(x · c) = (κq,z(x) · z + c) mod q The above lemma gives us an efficient algorithm for computing κq,z(x): start from κq,z(ϵ) = 0 (where ϵ is the empty string) and append the characters of x one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Even better: using a similar idea, we can concatenate the sketches of two strings in logarithmic time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let us denote with x·y the concatenation of the two string x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For this to work, our sketch should also remember the string’s length (only an additional logarithmic number of bits): the sketch of x becomes the pair (κq,z(x), |x|), where |x| denotes the number of characters in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For simplicity, in the following we will omit the string’s length (but assume we store it in the sketch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' It is easy to see that: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' κq,z(x · y) = (κq,z(x) · z|y| + κq,z(y)) mod q The length of the resulting string is, of course, |x · y| = |x| + |y|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Even if it appears that the above formula can be computed in constant time, the quantity z|y| mod q actually requires O(log |y|) time to be computed (by means of the fast exponentiation algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 3 We mention an additional crucial property of Rabin’s hashing: if x ̸= y, then κq,z(x) ̸= κq,z(y) with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This is implied by the following lemma: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let x ̸= y, with max(|x|, |y|) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then: P(κq,z(x) = κq,z(y)) ≤ n/q Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note that P(κq,z(x) = κq,z(y)) = P(κq,z(x)−κq,z(y) ≡q 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Now, the quantity κq,z(x)−κq,z(y) is, itself, a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let x − y be the string such that (x − y)[i] = x[i] − y[i] mod q, where we left-pad with zeros the shortest of the two strings (so that both have n characters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then, it is easy to see that: κq,z(x) − κq,z(y) mod q = κq,z(x − y) It follows that the above probability is equal to P(κq,z(x − y) ≡q 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Since x ̸= y, κq,z(x − y) is a polynomial of degree at most n over Zq (evaluated in z) and it is not the zero polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Recall that any non-zero univariate polynomial of degree n over a field has at most n roots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Since q is prime, Zq is a field and thus there are at most n values of z such that κq,z(x − y) ≡q 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Since we pick z uniformly from [0, q), the probability of picking a root is at most n/q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Choose a prime nc+1 ≤ q ≤ 2 · nc+1 for an arbitrarily large constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then, |κq,z(x)| ∈ O(log n) bits and, for any x ̸= y: P(κq,z(x) = κq,z(y)) ≤ n−c that is, x and y collide with low (inverse polynomial) probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 1Michael O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Rabin (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Fingerprinting by Random Polynomials 2Another variant of Rabin’s hashing draws a uniform prime q instead, and fixes z = |Σ| 3An alternative solution is to pre-compute all powers zi mod q, for 1 ≤ i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This, however, requires O(n) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 27 Later in these notes, Rabin fingerprinting will be used to solve pattern matching in the streaming model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' As noted above, this framework can be used also to sketch sets of integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note that, in this case, the sketch can be efficiently updated also when a new element is inserted in the set (provided that the element was not in the set before).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2 Sketching for Jaccard similarity - MinHash MinHash is a sketching algorithm used to estimate the similarity of sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' It was invented by Andrei Broder in 1997 and initially used in the AltaVista search engine to detect duplicate web pages and eliminate them from search results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Here we report just a definition and analysis of MinHash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For more details and applications see Leskovec et al.’s book [21], Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1 - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' MinHash is a technique for estimating the Jaccard similarity J(A, B) of two sets A and B: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1 (Jaccard similarity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' J(A, B) = |A∩B| |A∪B| Without loss of generality, we may assume that we work with sets of integers from the universe [1, n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This is not too restrictive: for example, if we represent a document as the set of all substrings of some length k appearing in it, we can convert those strings to integers using Rabin’s hashing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2 (MinHash hash function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let h be a hash function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The MinHash hash function of a set A is defined as ˆh(A) = min{h(x) : x ∈ A}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' it is the minimum of h over all elements of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3 (MinHash estimator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let ˆJh(A, B) be the indicator R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' defined as follows: ˆJh(A, B) = � 1 if ˆh(A) = ˆh(B) 0 otherwise (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1) Note that ˆJh(A, B) is a Bernoullian R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We prove the following remarkable property: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' If h : [1, n] → [1, n] is a uniform permutation, then E[ ˆJh(A, B)] = J(A, B) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let |A∪B| = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For i ∈ A∪B, consider the event smallest(i) = (∀j ∈ A∪B−{i})(h(i) < h(j)), stating that i is the element of A ∪ B mapped to the smallest hash h(i) (among all elements of A ∪ B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Since h is a permutation, exactly one element from A ∪ B will be mapped to the smallest hash (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' smallest(i) is true for exactly one i ∈ A ∪ B), so {smallest(i)}i∈A∪B is a partition of cardinality N = |A ∪ B| of the event space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Moreover, the fact that h is completely uniform implies that P(smallest(i)) = P(smallest(j)) for all i, j ∈ A∪B: every element of A∪B has the same chance to be mapped to the smallest hash (among elements of A∪B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This implies that P(smallest(i)) = 1/N for every i ∈ A ∪ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note that, if we know that smallest(i) is true and i ∈ A ∩ B, then ˆJh(A, B) = 1 (because i belongs to both A and B and h reaches its minimum min on i, thus ˆh(A) = ˆh(B) = min).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' On the other hand, if we know that smallest(i) is true and i ∈ (A ∪ B) − (A ∩ B), then ˆJh(A, B) = 0 (because i belongs to either A or B — not both — and h reaches its minimum min on i, thus either ˆh(A) ̸= ˆh(B) = min or min = ˆh(A) ̸= ˆh(B) holds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Using this observation and applying the law of total expectation (Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='11) to the partition {smallest(i)}i∈A∪B of the event space we obtain: 28 E[ ˆJh(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' B)] = � i∈A∪B P(smallest(i)) · E[ ˆJh(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' B) | smallest(i)] = � i∈A∪B 1 N · E[ ˆJh(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' B) | smallest(i)] = � i∈A∩B 1 N · E[ ˆJh(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' B) | smallest(i)] + � i∈(A∪B)−(A∩B) 1 N · E[ ˆJh(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' B) | smallest(i)] = � i∈A∩B 1 N · 1 + � i∈(A∪B)−(A∩B) 1 N · 0 = 1 N · � i∈A∩B 1 = 1 N |A ∩ B| = |A∩B| |A∪B| = J(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' B) The above lemma states that ˆJh(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' B) is an unbiased estimator for the Jaccard similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note that evaluating the estimator only requires knowledge of ˆh(A) and ˆh(B): an entire set is squeezed down to just one integer!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1 Min-wise independent permutations The main drawback of the previous approach is that h is a random permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' There are n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' random permutations of [1, n], so h requires log2(n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=') ∈ Θ(n log n) bits to be stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' What property of h makes Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4 go through?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' It turns out that we need the following: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='5 (Min-wise independent hashing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let h : [1, n] → [0, M) be a function from some family H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For any subset A ⊆ [0, M) and i ∈ A, let smallesth(A, i) = (∀j ∈ A − {i})(h(i) < h(j)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The family H is said to be min-wise independent if, for a uniform h ∈ H, P(smallesth(A, i)) = 1/|A| for any A ⊆ [1, n] and i ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In other words, H is min-wise independent if, for any subset of the domain, any element is equally likely to be the minimum (through a uniform h ∈ H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The definition could be made more general by further relaxing the uniformity requirement on h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Unfortunately, Broder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' [4] proved that any family of min-wise independent permutations must include at least en−o(n) permutations, so a min-wise independent function requires at least n log2 e ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='44n bits to be stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This lower bound is easy to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' First, observe that any h ∈ H identifies exactly one minimum in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Since every i ∈ A should have the same probability to be mapped to the minimum through a uniform h ∈ H, it follows that |A| must necessarily divide |H|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This should hold for every A ⊆ [1, n], so each k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , n should divide |H| and therefore |H| cannot be smaller than the least common multiple of all numbers 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The claim follows from the fact that lcm(1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , n) = en−o(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 4 There are two solutions to this problem: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' (k-min-wise independent hashing) We require P(smallesth(A, i)) = 1/|A| only for sets of cardi- nality |A| ≤ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 4Proof sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' lcm{1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , n} = � pi≤n pdi i , where the product runs over all prime numbers pi ≤ n and di is the largest integer such that pdi i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' It is easy to see that pdi i ≥ √n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' If pi ≥ √n, we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' If n1/3 ≤ pi < n1/2, then √n ≤ n2/3 ≤ p2 i ≤ pdi i (the last inequality follows from p2 i < n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In general, if n1/(j+1) ≤ pi < n1/j for some integer j ≥ 2, then √n ≤ nj/(j+1) ≤ pj i ≤ pdi i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We conclude lcm{1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , n} ≥ � pi≤n √n ≥ nn/(2 ln n) (the last step by the Prime Number Theorem - omitting low-order terms for simplicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Finally, since n = eln n, we obtain lcm{1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , n} ≥ en/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' A more precise calculation yields the stronger bound en−o(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' See https://en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='org/wiki/Chebyshev_function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 29 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' (Approximate min-wise hashing): we require P(smallesth(A, i)) = (1 ± ϵ)/|A| for a small error ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Also combinations of (1) and (2) are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' A hash with property (1) can be stored in O(k) bits of space and is a good compromise: in practice, k is the cardinality of the union of the two largest sets in our dataset (much smaller than the universe’s size n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' As far as solution (2) is concerned, there exist hash functions of size Θ(log(1/ϵ) · log n) bits with this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Such functions can be used to estimate the Jaccard similarity with absolute error ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For more details, see [19, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2 Reducing the variance The R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' ˆJh(A, B) of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3 is not a good estimator since it is a Bernoullian R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' and thus has a large variance: in the worst case (J(A, B) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='5), we have V ar[ ˆJh(A, B)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='25 and thus the expected error (standard deviation) of ˆJh(A, B) is � V ar[ ˆJh(A, B)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This means that on expectation (in the worst case) we are off by 50% from the true value of J(A, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We know how to solve this issue: just take the average of k independent such estimators, for sufficiently large k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let hi : [1, n] → [1, n], with i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , k, be k independent uniform permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We define the MinHash sketch of a set A to be the k-tuple: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='6 (MinHash sketch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' hmin(A) = (ˆh1(A), ˆh2(A), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , ˆhk(A)) In other words: the i-th element of hmin(A) is the smallest hash hi(x), for x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note that the MinHash sketch of a set A can be easily computed in O(k|A|) time, provided that h can be evaluated in constant time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then, we estimate J(A, B) using the following estimator: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='7 (Improved MinHash estimator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' J+(A, B) = 1 k k � i=1 ˆJhi(A, B) In other words, we compute the average of ˆJhi(A, B) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note that the improved MinHash estimator can be computed in O(k) time given the MinHash sketches of two sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We can immediately apply the double-sided additive Chernoff-Hoeffding bound (Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='17) and obtain that P(|J+(A, B) − J(A, B)| ≥ ϵ) ≤ 2e−ϵ2k/2 for any desired absolute error 0 < ϵ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Fix now any desired failure probability 0 < δ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' By solving 2e−ϵ2k/2 = δ we obtain k = 2 ln(2/δ)/ϵ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We can finally state: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Fix any desired absolute error 0 < ϵ ≤ 1 and failure probability 0 < δ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' By using k = 2 ϵ2 ln(2/δ) ∈ O � log(1/δ) ϵ2 � hash functions, the estimator J+(A, B) exceeds absolute error ϵ with probability at most δ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' P(|J+(A, B) − J(A, B)| ≥ ϵ) ≤ δ To summarize, we can squeeze down any subset of [1, n] to a MinHash sketch of O � log(1/δ) ϵ2 log n � bits so that, later, in O � log(1/δ) ϵ2 � time we can estimate the Jaccard similarity between any pair of sets (represented with MinHash sketches) with arbitrarily small absolute error ϵ and arbitrarily small failure probability δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note that it is easy to combine the MinHash sketches of two sets A and B so to obtain the MinHash sketch of A ∪ B (similarly, to compute the MinHash sketch of A ∪ {x} given the MinHash sketch of A): hmin(A ∪ B) = (min{ˆh1(A), ˆh1(B)}, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , min{ˆhk(A), ˆhk(B)}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 30 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3 Other metrics In general, a sketching scheme can be devised for most distance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' A distance metric over a set A is a function d : A × A → R with the following properties: Non-negativity: d(x, y) ≥ 0 Identity: d(x, y) = 0 iff x = y Simmetry: d(x, y) = d(y, x) Triangle inequality: d(x, z) ≤ d(x, y) + d(y, z) For example, the Jaccard distance dJ(x, y) = 1−J(x, y) defined over sets is indeed a distance metric (exercise: prove it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Of course, the sketching mechanism we devised for Jaccard similarity works for Jaccard distance without modifications (just invert the definition of the estimator of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Some examples of distances among vectors x, y ∈ Rd are: Lp norm (or Minkowski distance): Lp(x, y) = ��d i=1 |xi − yi|p�1/p L2 norm (or Euclidean distance): L2(x, y) = ��d i=1(xi − yi)2 L1 norm (or Manhattan distance): L1(x, y) = �d i=1 |xi − yi| L∞ norm: L∞(x, y) = max{|x1 − y1|, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , |xd − yd|} Cosine distance: dcos(x, y) = 1 − cos(x, y) = 1 − x·y ∥x∥·∥y∥ = 1 − �d i=1 xiyi √�d i=1 x2 i ·√�d i=1 y2 i Between strings, we have: Hamming distance between two equal-length strings: H(s1, s2) is the number of positions s1[i] ̸= s2[i] in which the two strings differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' On alphabet {0, 1} it is equal to L1(s1, s2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Edit distance between any two strings: Ed(s1, s2) is the minimum number of edits (substitutions, single-character inserts/deletes) that have to be applied to s1 in order to convert it into s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1 Sketching for Hamming distance We devise a simple sketching mechanism for Hamming distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Given two strings x, y ∈ Σn of the same length n, the Hamming distance dH(x, y) is the number of positions where x and y differ: dH(x, y) = n � i=1 (x[i] ̸= y[i]) where (x[i] ̸= y[i]) = 1 if x[i] ̸= y[i], and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Since dH is defined for strings of the same length n, we may scale it to a real number in [0, 1] as follows: d′ H(x, y) = dH(x, y)/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note that two strings are equal if and only if d′ H(x, y) = 0 (d′ H(x, y) is indeed a metric, see next section).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' If n is large, also the Hamming distance admits a simple sketching mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Choose a uniform i ∈ [1, n] and define hi(x) = x[i] Then, it is easy to see that P(hi(x) ̸= hi(y)) = d′ H(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Similarly to the Jaccard case, we can define an indicator R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' ˆHi(x, y) = � 1 if hi(x) ̸= hi(y) 0 otherwise (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2) 31 and obtain E[ ˆHi(x, y)] = d′ H(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Again, this indicator is Bernoullian and has a large variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' To reduce the variance, we can pick k uniform indices i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , ik ∈ [1, n] and define an improved indicator: H+(x, y) = 1 k k � j=1 ˆHij(x, y) Applying Chernoff-Hoeffding: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Fix an absolute error 0 ≤ ϵ ≤ 1 and failure probability 0 < δ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' By using k = 2 ϵ2 ln(2/δ) ∈ O � log(1/δ) ϵ2 � hash functions, the estimator H+(x, y) exceeds absolute error ϵ with probability at most δ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' P(|H+(x, y) − d′ H(x, y)| ≥ ϵ) ≤ δ Note that this sketch has a drawback: the family of hash functions {hi | 1 ≤ i ≤ n} contains only n elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This number is much smaller than the n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' permutations of the Jaccard case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' While this is not a problem here, it will become a problem in the next subsection (LSH), where a large supply of hash functions is essential in order to obtain a good locality-sensitive hash function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4 Locality-sensitive hashing (LSH) Locality-sensitive hash functions are used to accelerate the search of similar elements in a data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Similarity is usually measured in terms of a distance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' See Leskovec et al.’s book [21], Sections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4 - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='8 for applications of LSH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1 The theory of LSH Suppose our task is to find all similar pairs of elements (small d(x, y)) in a data set A ⊆ U (U is some universe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' While a distance-preserving sketch (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' for Jaccard distance) speeds up the computation of d(x, y), we still need to compute |A|2 distances in order to find all similar pairs!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' On big data sets this is clearly not feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' A locality-sensitive hash function for some distance metric d : U × U → R is a function h : U → [0, M) such that similar elements (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' d(x, y) is small) are likely to collide: h(x) = h(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This is useful to drastically reduce the search space with the following algorithm: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Scan the data set A and put each element x ∈ A in bucket H[h(x)] of a hash table H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Compute distances only between pairs inside each bucket H[i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Classic hash data structures use O(m) space for representing a set of m elements and support insertions and lookups in O(1) expected time (see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' More advanced data structures5 support queries in O(1) worst-case time with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In the following, we will therefore assume constant-time operations for our hash data structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' LSH works by first defining a distance threshold t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Ideally, we would like the collision probability to be equal to 0 for pairs such that d(x, y) > t and equal to 1 for pairs such that d(x, y) ≤ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For example, using a distance d : U × U → [0, 1] (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Jaccard distance) the ideal LSH function should be the one depicted in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In practice, we are happy with a good approximation: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' A (d1, d2, p1, p2)-sensitive family H of hash functions is such that, for a uniformly- chosen g ∈ H, we have: 5Dietzfelbinger, Martin, and Friedhelm Meyer auf der Heide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' “A new universal class of hash functions and dy- namic hashing in real time.” International Colloquium on Automata, Languages, and Programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Springer, Berlin, Heidelberg, 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 32 distance collision probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='0 collision probability vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' distance Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1: The ideal locality-sensitive hash function: elements whose distance is below the threshold t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='8 collide with probability 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' elements whose distance is above the threshold do not collide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' If d(x, y) ≤ d1, then P(g(x) = g(y)) ≥ p1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' If d(x, y) ≥ d2, then P(g(x) = g(y)) ≤ p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Intuitively, we want d1 and d2 to be as close as possible (d1 ≤ d2), p1 as large as possible, and p2 as small as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' To abbreviate, in the following we will say that h is a (d1, d2, p1, p2)-sensitive hash function when it is uniformly drawn from a (d1, d2, p1, p2)-sensitive family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For example, Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3 shows the behaviour of a (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='999, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='007)-sensitive hash function for Jaccard distance (see next subsection for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We now show how locality-sensitive hash functions can be amplified in order to obtain different (better) parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' AND construction Suppose H is a (d1, d2, p1, p2)-sensitive family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Pick uniformly r independent hash functions h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , hr ∈ H, and define: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2 (AND construction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' hAND(x) = (h1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , hr(x)) Then, if two elements x, y ∈ U collide with probability p using any of the hi, now they collide with probability pr using hAND (because the hi are independent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In other words, the curve becomes P(collision) = pr and we conclude: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' hAND is a (d1, d2, pr 1, pr 2)-sensitive hash function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Observe that, if the output of h is one integer, then hAND outputs r integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' However, we may use one additional collision-free hash function h′ to reduce this size to one integer: x is mapped to y = h′(hAND(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This is important, since later we will need to insert y in a hash table (this trick reduces the space by a factor of r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' OR construction Suppose H is a (d1, d2, p1, p2)-sensitive family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Pick uniformly b independent hash functions h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , hb ∈ H, and define: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4 (OR construction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We say that x and y collide iff hi(x) = hi(y) for at least one 1 ≤ i ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 33 Note: the OR construction can be simulated by simply keeping b hash tables H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , Hb, and inserting x in bucket Hi[hi(x)] for each 1 ≤ i ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then, two elements collide iff they end up in the same bucket in at least one hash table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Suppose two elements x, y ∈ U collide with probability p using any hash function hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then: For a fixed i, we have that P(hi(x) ̸= hi(y)) = 1 − p The probability that all hashes do not collide is P(∧b i=1hi(x) ̸= hi(y)) = (1 − p)b The probability that at least one hash collides is P(∨b i=1hi(x) = hi(y)) = 1 − P(∧b i=1hi(x) ̸= hi(y)) = 1 − (1 − p)b We conclude that the OR construction yields a curve of the form P(collision) = 1 − (1 − p)b so: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The OR construction yields a (d1, d2, 1−(1−p1)b, 1−(1−p2)b)-sensitive hash function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Combining AND+OR By combining the two constructions, each x is hashed through rb hash functions: we keep b hash tables and insert each x ∈ U in buckets Hi[hAND i (x)] for each 1 ≤ i ≤ b, where hAND i is the combination of r independent hash values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We obtain: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' If H is a (d1, d2, p1, p2)-sensitive family, then the AND+OR constructions with pa- rameters r and b yields a (d1, d2, 1 − (1 − pr 1)b, 1 − (1 − pr 2)b)-sensitive family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' It turns out (see next subsections) that by playing with parameters r and b we can obtain a function as close as we wish to the ideal LSH of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2 LSH for Jaccard distance Let ˆh be the MinHash function of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2 we have established that P(ˆh(A) = ˆh(B)) = J(A, B), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' the probability that two elements collide through ˆh is exactly their Jaccard similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Recall that we have defined the Jaccard distance (a metric) to be dJ(A, B) = 1 − J(A, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' But then, P(ˆh(A) = ˆh(B)) = 1 − dJ(A, B) and we obtain that ˆh is a (d1, d2, 1 − d1, 1 − d2)-sensitive hash function for any 0 ≤ d1 ≤ d2 ≤ 1, see Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Jaccard distance collision probability (MinHash function) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='0 collision probability (MinHash function) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Jaccard distance Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2: The MinHash function ˆh of Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2 is a (d1, d2, 1 − d1, 1 − d2)-sensitive function for any 0 ≤ d1 ≤ d2 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 34 Using the AND+OR construction, we can amplify ˆh and obtain a (d1, d2, 1 − (1 − (1 − d1)r)b, 1 − (1 − (1 − d2)r)b)-sensitive function for any 0 ≤ d1 ≤ d2 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For example, with r = 10 and b = 1200 we obtain a function whose behaviour is depicted in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Jaccard distance collision probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='0 collision probability vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Jaccard distance Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3: A (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='999, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='007)-sensitive hash function for Jaccard distance built with AND+OR construction with parameters r = 10 and b = 1200 starting from a (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3)-sensitive LSH function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Equivalently, we can take two closer points d1 and d2 on the curve: for example, this function is also (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='69, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='12)-sensitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The shape of the s-curve is dictated by the parameters b and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' As it turns out, b controls the steepness of the slope, that is, the distance between the two points where the probability becomes close to 0 and close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The larger b, the steeper the s-curve is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In other words, b controls the distance between d1 and d2 in our LSH: we want b to be large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Parameter r, on the other hand, controls the position of the slope (the point where the curve begins to decrease).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let p be the collision probability and dJ be the Jaccard distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The s-curve follows the equation p = 1−(1−(1−dJ)r)b By observing that the center of the slope is approximately around p = 1/2, one can determine the parameters b and r as a function of the slope position dJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let’s solve the following equation as a function of r: 1 − (1 − (1 − dJ)r)b = 1/2 We obtain (note that r should be an integer so we must approximate somehow): r = � ln � 1 − 2−1/b� ln(1 − dJ) � The fact that we have to approximate r to an integer means that the slope of the resulting curve will not be centered exactly at dJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' By playing with parameter b, one can further adjust the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Suppose we want to build a LSH to identify sets with Jaccard distance at most 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We choose a large b = 100000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then, the above equation gives us r = � ln(1−2−1/100000) ln(1−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='9) � = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Using these parameters, we obtain the LSH shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For example, one can extract two data points from this curve and see that this is a (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='85, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='95, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='99949, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='03076)-sensitive function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Clearly, a large b has a cost: in Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='7, we have to compute r·b = 5·105 MinHash functions for each set, which means that we have to apply 5 · 105 basic hash functions h (see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2) to each element of each set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Letting t = r · b, this translates to O(|A| · t) running time for a set A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Dahlgaard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' [9] improved this running time to O(|A| + t log t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Another solution is to observe that 35 Jaccard distance collision probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='0 collision probability vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Jaccard distance Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4: LSH built for Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' the t MinHashes are completely independent, thus their computation can be parallelized optimally (for example, with a MapReduce job running over a large cluster).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Observe also that a large value of b requires a large family of hash functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' While this is not a problem with the Jaccard distance (where the supply of n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' permutations is essentially unlimited), it could be a problem with the sketch for Hamming distance presented in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' There, we could choose only among n hash functions, n being the strings’ length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' It follows that the resulting LSH scheme is not good for small strings (small n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3 Nearest neighbour search One application of LSH is nearest neighbour search: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='8 (Nearest neighbour search (NNS)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For a given distance threshold D, preprocess a data set A of size |A| = n in a data structure such that later, given any data point x, we can quickly find a point y ∈ A such that d(x, y) ≤ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' To solve the NNS problem, let H be a (D′, D, p1, p2)-sensitive family, with D′ as close as possible to (and smaller than) D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Suppose moreover that h(x) can be evaluated in time th (this time is proportional to the size/cardinality of x) and d(x, y) can be computed in time td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note that td can be reduced considerably by employing sketches — see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We amplify H with an AND+OR construction with parameters r (AND) and b (OR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Our data structure is formed by b hash tables H1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , Hb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For each of the n data points x ∈ A, we compute the b functions hAND i (x) in total time O(n · b · r · th) and insert in Hi[hAND i (x)] a pointer to the original data point x (or to its sketch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Assuming that a hash table storing m pointers occupies O(m) words of space and can be constructed in (expected) O(m) time, we obtain: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Our NNS data structure can be constructed in O(n·b·r ·th) time and occupies O(n·b) space (in addition to the original data points — or their sketches).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' To answer a query x, note that we are interested in finding just one point y such that d(x, y) ≤ D: we can stop our search as soon as we find one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In O(th · b · r) time we compute the hashes hAND i (x) for all 1 ≤ i ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In the worst case, all the n data points y are such that d(x, y) > D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The probability that one such point ends up in bucket Hi[hAND i (x)] is at most pr 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' As a result, the expected number of false positives in each bucket Hi[hAND i (x)] is at most n · pr 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' in total, this yields n · b · pr 2 false positives that need to be checked against x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For each of these false positives, we need to compute a distance in time td.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We obtain: 36 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let: FP = n · b · pr 2 be the expected number of false positives in the worst case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' T = b · r be the total number of independent hash functions used by our structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Our NNS data structure answers a query in expected time O(th·T +FP ·td).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' If there exists a point within distance at most D′ from our query, then we return an answer with probability at least 1 − (1 − pr 1)b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Consider the (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='999, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='007)-sensitive family of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This function has been built with AND+OR construction with parameters r = 10 and b = 1200 taking as starting point the (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3)-sensitive hash function of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3 (in fact, 1 − (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='6r)b ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='999 and 1 − (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3r)b ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We can therefore use this hash to solve the NNS problem with threshold D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='10 states that at most FP = n · b · pr 2 ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='007 · n false positives need to be explicitly checked against our query (compare this with a naive strategy that compares 100% of the n points with the query).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Moreover, if at least one point within distance D′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4 from our query exists, we will return a point within distance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='7 with probability at least 1 − (1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='6r)b ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The data structure uses space proportional to b = 1200 words (a few kilobytes) for each data point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' note that, in big data scenarios, each data point (for example, a document) is likely to use much more space than that so this extra space is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 37 Chapter 4 Mining data streams A data stream is a sequence x = x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , xm of elements (without loss of generality, integers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We receive these elements one at a time, from x1 to xm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Typically, m is too large and we cannot keep all the stream in memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The goal of streaming algorithms is to compute interesting statistics on the stream while using as little memory as possible (usually, poly-logarithmic in m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' A streaming algorithm is evaluated on these parameters: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Memory used (as a function of, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=', m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Delay per element: the worst-case time taken by the algorithm to process each stream element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Probability of obtaining a correct solution or a good approximation of the correct result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Approximation ratio (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' the value returned by the algorithm is a (1 ± ϵ) approximation of the correct answer, for a small ϵ ≥ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' A nice introduction to data sketching and streaming is given in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1 Pattern matching The first example of stream statistic we consider is pattern matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Say the elements xi belong to some alphabet Σ: the stream is a string of length m over Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Suppose we are given a pattern y = y1y2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' yn ∈ Σn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The pattern’s length n is smaller than m, but also n could be very large (so that y too does not fit in memory or cache).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The question we tackle in this section is: how many times does y appear in x as a substring y = xixi+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' xi+n−1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1 (Intrusion Detection and Prevention Systems (IDPSs)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' IDPSs are software tools that scan network traffic in search of known patterns such as virus fragments or malicious code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The searched patterns are usually very numerous, so the memory usage and delay of the used pattern matching algorithm is critical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Ideally, the algorithm should work entirely in cache in order to achieve the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' See also the paper [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1 Karp-Rabin’s algorithm Rabin’s hashing is the main tool we will use to solve the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' First, we note that the technique itself yields a straightforward solution, even though in O(n) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In the next section we refine this solution to use O(log n) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Suppose we have processed the stream up to x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , xi (i ≥ n) and that we know the hash values κq,z(xi−n+1xi−n+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' xi) and κq,z(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' By simply comparing these two hash values (in constant time) 38 we can discover whether or not the patter occurs in the last n stream’s characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The crucial step is to update the hash of the stream when a new element xi+1 arrives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This is not too hard: we have to subtract character xi−n+1 from the stream’s hash and add the new character xi+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This can be achieved as follows: κq,z(xi−n+2xi−n+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' xi+1) = (κq,z(xi−n+1xi−n+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' xi) − xi−n+1 · zn−1) · z + xi+1 mod q The value zn−1 mod q can be pre-computed, so the above operation takes constant time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note that, since we need to access character xi−n+1, at any time the algorithm must keep the last n characters seen in the stream, thereby using O(n) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Analysis Note that, if there are no collisions between the pattern and all the m − n + 1 ≤ m stream’s substrings of length n, then the algorithm returns the correct result (number of occurrences of the pattern in the stream).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' From Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1, the probability that the pattern collides with any of those substrings is at most n/q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' By union bound, the probability that the patter collides with at least one substring is mn/q ≤ m2/q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We want this to happen with small (inverse polynomial probability): this can be achieved by choosing a prime q in the range [mc+2, 2 · mc+2], for any constant c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Such a prime (and therefore the output of Rabin’s hash function) can be stored in O(log m) bits = O(1) words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We obtain: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The Karp-Rabin algorithm solves the pattern matching problem in the streaming model using O(n) words of memory and O(1) delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The correct solution is returned with high (inverse- polynomial) probability 1 − m−c, for any constant c ≥ 1 chosen at initialization time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' There exist also deterministic algorithms with O(1) delay and O(n) space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' However, as we show in the next section, Karp-Rabin’s randomization enables an exponentially more space-efficient solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2 Porat-Porat’s algorithm The big disadvantage of Karp-Rabin’s algorithm is that it uses too much memory: O(n) words per pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In this section we study an algorithm described by Benny Porat and Ely Porat in [25] that uses just O(log n) words of space and has O(log n) delay per stream’s character 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Other algorithms are able to reduce the delay to the optimal O(1) (see [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For simplicity, assume that n is a power of two: n = 2e for some e ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The algorithm can be generalized to any n in a straightforward way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The overall idea is to: Keep a counter occ initialized to 0, storing the number of occurrences of y found in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Keep the hashes of all 1 + e = 1 + log2 n prefixes of y whose length is a power of two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Keep the occurrences of those prefixes of y on the stream, working in e levels: level 0 ≤ i < e records all occurrences of the prefix y[1, 2i] in a window containing the last 2i+1 stream’s characters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Using a clever argument based on string periodicity, show that this information can be ”compressed” in just O(1) space per level (O(log n) space in total).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Before the new character xj arrives: for every level i > 0, if position p = j − 2i+1 (the oldest position in the window of level i) was explicitly stored because it is an occurrence of y[1, 2i], then remove it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In such a case, check also if p is an occurrence of y[1, 2i+1] (do this check using fingerprints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' If this is the case, then insert p in level i + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' moreover, if i + 1 = e then we have found an occurrence of y: increment occ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 1Note that, no matter how large n is, O(log n) words will fit in cache.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' O(log n) delay in cache is by far more desirable than O(1) delay in RAM: the former is hundreds of times faster than the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 39 When the new character xj arrives: check if it is an occurrence of y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' If yes, push position j to level 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Moreover, (cleverly) update the hashes of all positions stored in all levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1 depicts two steps of the algorithm: before and after the arrival of a new stream character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Algorithm 1 implements one step of the above procedure (hiding details such as compression of the occurrences and update of the hashes, which are discussed below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The window at level i is indicated as Wi and it is a set of positions (integers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We assume that the stream is ended by a character # not appearing in the pattern (this is just a technical detail: by the way we define one update step, this is required to perform the checks one last time at the end of the stream).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Algorithm 1: new stream character(xj) foreach level i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , e − 1 do if j − 2i+1 ∈ Wi then Wi ← Wi − {j − 2i+1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' wi ← κq,z(x[j − 2i+1, j − 1]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' // hash of the full window 1 if κq,z(y[1, 2i+1]) == wi then 2 if i == e − 1 then 3 occ ← occ + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 4 else 5 Wi+1 ← Wi+1 ∪ {j − 2i+1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' if xj == y1 then W0 ← W0 ∪ {j};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Compressing the occurrences We have log n levels, however this is not sufficient to claim that the algorithm uses O(log n) space: in each level i, there could be up to 2i occurrences of the pattern’s prefix y[1, 2i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In this paragraph we show that all the occurrences in a window can be compressed in just O(1) words of space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The key observation is that, in each level, we store occurrences of the pattern’s prefix of length K = 2i in a window of size 2K = 2i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Now, if there are at least three such occurrences, then at least two of them must overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' But these are occurrences of the same string y[1, 2i], so if they overlap the string must be periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Finally, if the string is periodic then all its occurrences in the window must be equally-spaced: we have an occurrence every p positions, for some integer p (a period of the string).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then, all occurrences in the window can be stored in just O(1) space: just remember the first occurrence r1, the period p, and the number t of occurrences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This representation is also easy to update (in constant time) upon insertion of new occurrences to the right (which must follow the same rule) and removal of an occurrence to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We now formalize this reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3 (Period of a string).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let S be a string of length K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We say that S has period p if and only if S[i] = S[i + p] for all 1 ≤ p ≤ K − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The string S = abcabcabcabca, of length K = 13, has periods 3, 6, 9, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='5 (Wilf’s theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Any string having periods p, q and length at least p + q − gcd(p, q) also has gcd(p, q) as a period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 40 a b b a a b b a a b b a a b a b b Level 0 Level 1 Level 2 Level 3 Stream b a a b b a a b b a a b a b b a Pattern a b b a a b b a a b b a a b a b b a Level 0 Level 1 Level 2 Level 3 Stream b a a b b a a b b a a b a b b a Pattern Pattern occurrence!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' (A) (B) ✓ Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1: Each colored dot at level i represents an occurrence of a prefix of length 2i of the pattern (underlined with corresponding color).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' (A) We have two such occurrences at level 0, one occurrence at levels 1 and 2, and two occurrences at level 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Some of these occurrences are candidates that could be promoted to the next level: these are the leftmost occurrences at levels 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Unfortunately, none of these occurrences can be promoted since they are not occurrences of a prefix of length 2i+1 of the pattern: the leftmost occurrence at level 0 is not an occurrence of ”ba” and the occurrence at level 1 is not an occurrence of ”baab”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' They will therefore be eliminated before the next stream’s character arrives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' (B) A new stream character (”a”, in bold) has arrived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' All occurrences, except the eliminated ones, have been shifted to the left in the levels’ windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Now, three occurrences are candidates that could be promoted to the next level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The occurrence at level 0 is indeed an occurrence of ”ba”, therefore it will be promoted to level 1 before the next stream’s character arrives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The occurrence at level 2 is not an occurrence of ”baabbaab”, therefore it will be eliminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Finally, the leftmost occurrence at level 3 is an occurrence of the full pattern: before removing it from the window, we will increment the counter occ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 41 Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Consider the string above: S = abcabcabcabca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The string has periods 6, 9 (with gcd(6, 9) = 3) and has length 13 > 6 + 9 − 3 = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Wilf’s theorem can be used to deduce that the string must also have period gcd(6, 9) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Wilf’s theorem can be used to prove the following (exercise): Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let P be a string of length K, and S be a string of length 2K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' If P occurs in S at positions r1 < r2 < · · · < rq, with q ≥ 3, then rj+1 = rj + p, where p = r2 − r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The lemma provides a compressed representation for all the occurrences in a window: just record (r1, p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This representation is easy to update in constant time whenever an occurrence exits/enters the window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Updating the fingerprints The last thing to show is how to efficiently compute wi = κq,z(x[j − 2i+1, j − 1]) at level i (needed at Line 1 of the algorithm), that is, the fingerprint of the whole window when the first occurrence stands at the beginning of the window: r1 = j − 2i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Consider the window Wi at level i, and the two smallest positions r1, r2 ∈ Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let x = x[1, j − 1] be the current stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We keep in memory three fingerprints (see Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2): (A) κq,z(x): the fingerprint of the whole stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' (B) κq,z(x[r1, r2 − 1]): the fingerprint of the stream’s substring standing between r1 (included) and r2 (excluded), whenever Wi contains at least two positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' (C) κq,z(x[1, r1 − 1]): the fingerprint of the stream’s prefix ending at r1 − 1, whenever Wi contains at least one position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 xj r1 r2 A B C window Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='2: For each level (window), we keep three fingerprints: A (full stream, unique for all windows), B (string between first two pattern’s occurrences), and C (from beginning of the stream to the first pattern’s occurrence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Knowing A,B, and C we can easily compute the fingerprint wi of the whole window when r1 = j − 2i+1: wi = A − C · z2i+1 mod q Note that z2i+1 mod q can easily be pre-computed for any i ≤ log n at the beginning of the algorithm using the recurrence z2i+1 = (z2i)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We now show how to update the three fingerprints A, B, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Updating A Fingerprint A - the full stream - can be updated very easily in constant time each time a new stream character arrives (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 42 Updating B - case 1 B needs to be updated in two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The first case happens when r2 enters in the window (before that, only r1 was in the window): see Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then, notice that x[r2, j − 1] = y[1, 2i], so we have the fingerprint D = κq,z(y[1, 2i]) = κq,z(x[r2, j − 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 xj r1 r2 B C window D A Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3: Updating B - case 1: r2 enters in the window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' It follows that B can be computed as: B = � A − D − C · z|D|+|B|� z−|D| mod q In the above equation, the quantity z|D|+|B| ≡q z2i+(r2−r1) can be computed one step at a time (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' multiplying z by itself 2i + (r2 − r1) times modulo q) while the stream characters between r1 and r2 + 2i − 1 arrive (constant time per character per level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The log n values z−|D| = z−2i mod q can be pre-computed before the stream arrives in O(log m) time as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' z2i+1 ≡q (z2i)2, and z−2i mod q can be computed in O(log q) = O(log m) time using the equality a−1 ≡q aq−2 and fast exponentiation: z−2i ≡q z2i·(q−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Updating B - case 2 The second case where we need to update B is when r1 exits the window and r3 is in the window: B should become the fingerprint of the string between r2 and r3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' See Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 xj r1 r2 r3 B C window B’ Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='4: Updating B - case 2: r1 exits the window and r3 is in the window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' It turns out that in this case nothing needs to be done: The new fingerprint is B′ = B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' To see this, note that (1) r3 − r2 = r2 − r1 by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='7, and (2) r1 and r2 are both occurrences of the same string of length 2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Since r2 − r1 ≤ 2i, then x[r1, r2 − 1] = x[r2, r3 − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Updating C - case 1 C needs to be updated in two cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The first case happens when r1 enters in the window (before that, the window was empty: Wi = ∅).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' See Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' As in case B1, notice that we have the fingerprint D = κq,z(y[1, 2i]) = κq,z(x[r1, j − 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then: C = (A − D) · z−|D| mod q 43 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 xj r1 A C window D Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='5: Updating C - case 1: r1 enters in the window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Updating C - case 2 The last case to consider is when r1 exits the window and r2 is in the window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' See Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16 x17 xj r1 r2 C’ B C window Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='6: Updating C - case 2: r1 exits the window and r2 is in the window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This is achieved as follows: C′ = C · z|B| + B mod q where z|B| ≡q zr2−r1 is computed as zr2−r1 ≡q z2i+(r2−r1) · z−2i (both computed as in case B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Final result Observe that each fingerprint update can be performed in constant time (per level, thus O(log n) time per stream’s character).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We obtain: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let m be the stream’s length and n ≤ m be the pattern’s length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Porat-Porat’s algorithm solves the pattern matching problem in the streaming model using O(log n) words of memory and O(log n) delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The correct solution is returned with high (inverse-polynomial) probability 1−m−c, for any constant c ≥ 1 chosen at initialization time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Breslauer and Galil in [3] reduced the delay to O(1) while still using O(log n) words of space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='3 Streamed approximate pattern matching We describe a modification of Porat-Porat’s algorithm that allows finding all stream occurrences xi,n = xi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' xi+n−1 of a pattern y = y1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' yn such that dH(xi,n, y) ≤ k for any parameter k, where dH is the Hamming distance between strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 44 For simplicity, we first describe the algorithm for k = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' zero or one mismatch between the pattern and the stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let yi:d = yi yi+d yi+2d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' In other words, yi:d is the sub-pattern built by extracting one every d characters from y, starting from character yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We call yi:d a shift of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Consider two strings x and y, both of the same length n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Clearly, x = y if and only if xi:d = yi:d for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Assume now that dH(x, y) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then, note that the error is captured by exactly one of the d shifts: there exists one i′ ∈ [1, d] such that xi′:d ̸= yi′:d, and xi:d = yi:d for all i ̸= i′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let x =abracadabra and y =abbacadabra, with dH(x, y) = 1 (the mismatch is underlined).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Pick d = 2 and consider the two shifts (per word) x1:2 = arcdba, x2:2 =baaar, y1:2 = abcdba, y2:2 =baaar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then: x1:2 ̸= y1:2 x2:2 = y2:2 What if dH(x, y) = k > 1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then, the number of shifts i′ such that xi′:d ̸= yi′:d could be smaller than k (but never larger).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Notice that this happens precisely when the distance |j′ − j| between two mismatches xj ̸= yj and xj′ ̸= yj′ is a multiple of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let x =abracadabra and y =abbacaaabra, with dH(x, y) = 2 (the two mismatches are underlined).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Pick d = 2 and consider the two shifts (per word) x1:2 = arcdba, x2:2 =baaar, y1:2 = abcaba, y2:2 =baaar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then: x1:2 ̸= y1:2 x2:2 = y2:2 In particular, the Hamming distance is 2 but only one of the two shifts generates a mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This happens because the two mismatches are distanced 4 positions, which is a multiple of d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' It is easy to see that the above issue does not happen if d does not divide the distance between the two mismatches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let x =abracadabra and y =abbacaaabra, with dH(x, y) = 2 (the two mismatches are underlined).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Pick d = 3 and consider the three shifts (per word) x1:3 =aadr, x2:3 =bcaa, x3:3 =rab, y1:3 =aaar, y2:3 =bcaa, y3:3 =bab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then: x1:3 ̸= y1:3 x2:3 = y2:3 x3:3 ̸= y3:3 Now, two shifts generates a mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This property can be summarized in a corollary: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let x, y be two words of length n, and consider their d shifts xi:d, yi:d for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then: If dH(x, y) = 0, then xi:d = yi:d for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' If dH(x, y) = 1, then xi:d ̸= yi:d for exactly one i ∈ [1, d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' If dH(x, y) > 1, then xi:d ̸= yi:d for at least two values i ∈ [1, d], provided that there exist two mismatches whose distance |j − j′| is not a multiple of d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 45 Consider the distance |j − j′| between (the positions of) any two mismatches between x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Consider moreover the smallest ⌈log2 n⌉ prime numbers P = {p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , p⌈log2 n⌉}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Clearly, |j − j′| cannot be a multiple of all numbers in P: this would imply that |j − j′| ≥ � p∈P p > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This immediately suggests an algorithm for pattern matching at Hamming distance at most 1 between two strings x, y of length n: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For each d ∈ P = {p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , p⌈log2 n⌉}, do the following: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' For each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , d, compare xi:d and yi:d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' If at least two values of i ∈ [1, d] are such that xi:d ̸= yi:d, then exit and output “dH(x, y) > 1”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' If step (2) never exits, then output “dH(x, y) ≤ 1”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' It is straightforward to adapt the above algorithm to the case where x is streamed and |x| = m > n: just break the stream into d sub-streams (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' xi xi+d xi+2d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , for all i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' , d), for every d ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' This allow implementing the comparisons xi:d ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='= yi:d using Porat-Porat’s algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Note that we run O(log2 n) parallel instances of Porat-Porat’s algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We obtain: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Let m be the stream’s length and n ≤ m be the pattern’s length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The above modifi- cation of Porat-Porat’s algorithm finds all occurrences of the pattern at Hamming distance at most 1 in the stream using O(log3 n) words of memory and O(log3 n) delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The correct solution is returned with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We can easily extend the above idea to k ≥ 1 mismatches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Assume that dH(x, y) > k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Consider any group of k + 1 mismatches between x and y, at positions i1 < · · · < ik+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' We want to find a prime number d ≥ k + 1 such that d does not divide ij − ij′, for all 1 ≤ j′ < j ≤ k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' Then, we are guaranteed that xi:d ̸= yi:d for at least k + 1 shifts i, since no pair of mismatches ij, ij′ can fall in the same shift (which would imply that d divides |ij − ij′|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6dAyT4oBgHgl3EQf2fn3/content/2301.00754v1.pdf'} +page_content=' The integer d does not divide ij − ij′ for all 1 ≤ j′ < j ≤ k + 1 if and only if d does not divide their product � 1≤j′