Part 1: Hashing and Its Many Applications

Transcription

1 1 Part 1: Hashing and Its Many Applications Sid C-K Chau

2 Why Randomized Algorithms? 2 Randomized Algorithms are algorithms that mae random choices during the execution We also mae lots of random choices everyday, because Lac of information Convenience and simplicity To diversify ris and try luc! These reasons apply to algorithmic design But unscrupulous random choices may end with useless results Question: How do we mae smart random choices? In practice: Simple random choices often wor amazingly well In theory: Simple maths can justify these simple random choices

3 Applications of Randomized Algorithms 3? Randomized algorithms are especially useful for applications with Large data set and insufficient memory Limited computational power Uninformed nowledge Minor fault tolerability A long list of applications include Information retrieval, databases, bioinformatics (e.g. Google, DNA matching) Networing, telecommunications (e.g. AT&T) Optimization, data prediction, financial trading Artificial intelligence, machine learning Graphics, multi-media, computer games Information security, and a lot more

4 A Key Example: Hashing 4 0A 084 ffe d3e string-based hashing for address boo Hashing enables large-scale, fast data processing Expedite the performance of large data/file systems in search engines (Google, Bing) Enable fast response time in small low-power devices (iphone, ipad) Hashing is a random sampling/projection of some more complicated data objects (e.g. strings, graphs, functions, sets, data structures) E.g. String-based hashing maps a input string to a shorter hash (string) by a hash function Assuming that a hash function is selected randomly (without a priori nowledge) from a large class of hash functions Hence, when we do not specify the detailed implementation of a particular hash function, the behaviour of hashing appears probabilistic

5 Balls and Bins Model 5 0A 084 ffe d3e n bins m balls A generic model for hashing is balls-and-bins model Throw m balls into n bins, such that each ball is uniformly randomly distributed among the bins Interpretations of the model Balls = data objects, Bins = hashes (Coupon Collector Problem) Balls = coupons, Bins = types of coupons (Birthday Attac Prob.) Balls = people, Bins = birthdates Key questions Efficiency: How many non-empty bins? Performance: What is the maximum number of balls in all the bins? Balls-and-bins model is a random model Its behaviour is naturally analysed by probability theory

6 Poisson Approximation 6 The probability that bin i has r balls follows binominal distribution P X i = r = m r 1 n r 1 1 n m r = 1 r! m m 1 m r+1 n r But the expression can be too unwieldy When m and n are very large, we can approximate by m m 1 m r+1 n r m n m Hence, P X i = r e n r and 1 1 r! n m r e m n This is nown as Poisson distribution Po(r) = e μ μ r m n r r! 1 1 n m r The mean of Poisson distribution is μ = m n The probability of a non-empty bin is P X i 0 1 Po 0 = 1 e μ Po(r) = e μ μ r r!

7 Maximum Load 7 Recall a well-nown technique called Union Bound P X 1 r or or X n r P X 1 r + + P X n r The probability that all bins have less than M balls is 1 P max i=1,,n X i M 1 n P X i M If M > μ = m n, then P X i M e μ e μ M (Shown by Chernoff Bound in homewor) If m = n (hence = 1) and M = P X i ln ln n e 1 e M M M e ln ln n = 3 ln n e M M ln ln n, then ln ln n ln ln n ln n 1 P max X e i 1 n i=1,,n ln ln n Therefore, the maximum load is larger than (i.e., P max X i 0, as n ) i=1,,n ln ln n Or we say that the maximum load is less than ln ln n = e (ln ln ln n ln ln n) ln ln n e ln ln n ln ln n ln ln n ln ln n) e(ln ln ln n e 1 1 en ln ln n has a vanishing probability with high probability.

8 B. Roberts K. G. Smith A. Williams C. Phillips D. Johnson Bloom filter n-bit string H. Morrison (Not a member) A. Clare (False positive) Bloom Filter Instead of hashing from a string, we also consider more complicated objects A Bloom filter maps a set of strings to a n-bit string There are hash functions, each hash function h maps a string to a value in {1,..,n} We initially set the Bloom filter to be an n-bit zero string If we include a string s in the Bloom filter, we set the h (s)-th bit in the Bloom filter to be one for every To validate whether a string s is a member of a Bloom filter, we chec if the h (s)-th bit in the Bloom filter is one for every A string belonging to a Bloom filter will be confirmed as a member by the validation (i.e., there is no false negative) However, it is possible that a string not belonging to a Bloom filter will also be confirmed as a member by the validation (i.e., there can be false positive) 8

9 Sid C-K Chau Applications of Bloom Filter Useful to applications with minor fault tolerance to false positives: Bloom filter is a compact representation of a set of strings 1) Spell and password checers with a set unsuitable words 2) Distributed database query 3) Content distribution and web cache 4) Peer-to-peer networs 5) Pacet filtering and measurement of pre-defined flows 6) Information security, computer graphics, etc. Distributed database query Peer-to-peer networs 9 C. Phillips B. Roberts K. G. Smith. A. Williams D. Johnson K. G. Smith C. Phillips B. Roberts K. G. Smith A. Williams D. Johnson K. G. Smith I want to now who has C. Edwards, K. Johnson, and? Do we have any common items?

10 Optimization of Bloom Filter 10 We want to minimize the number of false positives There are m strings to be included in an n-bit string Bloom filter There are hash functions, each hash function h maps a string to a value in {1,..,n} The probability that a particular bit in the Bloom filter becomes one after including m strings is n m 1 e m n, assuming that n and m are very large Consider validating if a random string is included in the Bloom filter or not The probability that the validation succeeds is n m 1 e m n f m,n () f m,n () is also the probability of a false positive. Hence, we want to minimize f m,n with respect to d ln f m,n () d = ln(1 e m/n ) + m n e m/n 1 e m/n Hence, d ln f m,n() = 0 = ln 2 n and f d m m,n = 1 2 = n/m For instance, if m =100 and f m,n = 0.01, then n = 938 and = 7

11 A stream of items with multiple occurrences d a a d d c d c c c b b b b b b b b a a a Heavy Hitter Problem Find the most frequent items in a stream In a networ, find the users who consume the most bandwidth by observing a stream of pacets In a search engine, find the most queried phrases From the transactions of a supermaret, find the most purchased items Heavy hitter problem There is a stream of items with multiple occurrences We want to find the items with the most occurrences, when observing the stream continuously We do not now the number of distinct items in a prior manner We are only allowed to use storage space much less than the number of items in the stream Algorithms that process a stream of data with tight space consumption are called streaming algorithms 11

12 rows of counters (hash funcs.) Sid C-K Chau Count-min Setch 12 d a a d c d b b m/ columns of counters We use an approach similar to the Bloom filter called count-min setch A setch is an array of m/ counters, *C i,j + There are hash functions, each hash function h i maps an item to a value in {1,..,m/} Initially set all counters to be zero (C i,j = 0) When we observe an item s in the stream, increase the h i (s)-th counter (C i,hi (s) = C i,hi (s)+1) for every i At the end, we obtain the number of occurrences of an item s by the minimum of all the counters that are mapped by s as N s = min *C i,hi (s): i = 1,, + N(s) is of course an overestimate of the true number of occurrences, because multiple items can be mapped to the same counter by a hash function However, N(s) is not far from the true value

13 Principle of Count-min Setch 13 Let the true number of occurrences of item s be T(s) Let the total number of occurrences of all items be T The probability that N(s) T(s)+ T is at most Let X t be the random item at time t=1,,t mε, where 1 T Then the counter C i,hi (s) = t=1 1,h i X t = h i s - and is a random variable, where 1, - is an indicator function We obtain the expected deviation of C i,hi (s) from T(s) by T E,C i,hi (s) T(s)- = E, t=1:x t s 1,h i X t = h i s -- T T = t=1:x t s E 1 h i X t = h i s = t=1:x t s P h i X t = h i s T P h i X t = h i s = T m Recall Marov inequality: P X x E X, for positive x x 0 P X < x + x P X x y y P X = y = E X Hence, P*C i,hi (s) T(s) εt+ E C i,h i s T(s) εt = εm Continue in the next slide

14 Principle of Count-min Setch 14 Follow from the last slide Since P*C i,hi (s) T(s) εt+, P* min *C εm i,h i=1,.., i (s)+ T s + εt+ If we minimize = m ε/e, εm εm with respect to, then = e mε/e, and P*N(s) T s + εt+ e mε/e If we let = ln 1 δ and m = ln 1 δ e ε, then P*N(s) T s + εt+ δ Therefore, is a tolerance threshold that bounds the deviation of N(s) from count-min setch, and is an error probability that bounds the probability of N(s) deviating for the at most T For example, if we set = 0.1 and = 0.01, then the number of counters we need is m = 125 and the number of hash functions is = 5 (note that both m and are independent of the number of items in the stream) Streaming algorithms can do much more powerful tass than finding the most frequent items, such as the distributions, correlations and other statistics in a stream of items in a continuous fashion εm

15 Summary 15 Randomized algorithms are algorithms that mae smart random choices during execution Hashing is a ey example that enables large-scale and fast data processing A simple balls-and-bins model can characterize the probabilistic properties of hashing (e.g. maximum load) A Bloom filter is an example that generates a hash to determine the membership of a set of strings Streaming algorithms use a random compact data structure (setches) to determine the statistics of a stream of items in continuous fashions Hashing can be regarded as a random projection from a high dimensional space of data to a low dimensional space of hashes

16 References 16 Main reference: Mitzenmacher and Upfal boo, Probability and Computing: Randomized Algorithms and Probabilistic Analysis Chapter : Balls-and-Bins model, Poisson distribution Chapter 5.5.4: Bloom filter Chapter 13.4: Count-min setch Additional references Broder and Mitzenmacher, Networ Applications of Bloom Filters: A Survey, Internet Mathematics 1 (4), pp Cormode and Hadjieleftheriou, Finding the frequent items in streams of data, Communications of the ACM, Oct 2009, pp More related materials are available at