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CS 473: Algorithms, Fall 2019 Universal and Perfect Hashing Lecture 10 September 26, 2019 Chandra and Michael (UIUC) cs473 1 Fall 2019 1 / 45 Today's lecture: Review pairwise independence and related constructions (Strongly) Universal hashing Perfect hashing Announcements and Overview Pset 4 released and due on Thursday, October 3 at 10am. Note one day extension over usual deadline. Midterm 1 is on Monday, Oct 7th from 7-9.30pm. More details and conflict exam information will be posted on Piazza. Next pset will be released after the midterm exam. Chandra and Michael (UIUC) cs473 2 Fall 2019 2 / 45 Announcements and Overview Pset 4 released and due on Thursday, October 3 at 10am. Note one day extension over usual deadline. Midterm 1 is on Monday, Oct 7th from 7-9.30pm. More details and conflict exam information will be posted on Piazza. Next pset will be released after the midterm exam. Today's lecture: Review pairwise independence and related constructions (Strongly) Universal hashing Perfect hashing Chandra and Michael (UIUC) cs473 2 Fall 2019 2 / 45 Part I Review Chandra and Michael (UIUC) cs473 3 Fall 2019 3 / 45 Pairwise independent random variables Definition Random variables X1; X2;:::; Xn from a range B are pairwise independent if for all 1 ≤ i < j ≤ n and for all b; b0 2 B, 0 0 Pr[Xi = b; Xj = b ] = Pr[Xi = b] · Pr[Xj = b ] : Chandra and Michael (UIUC) cs473 4 Fall 2019 4 / 45 Interesting case: n = m = p where p is a prime number Pick a; b uniformly at random from f0; 1; 2;:::; p − 1g Set Xi = ai + b Only need to store a; b. Can generate Xi from i. Relies on the fact that Zp = f0; 1; 2;:::; p − 1g is a field Constructing pairwise independent rvs Suppose we want to create n pairwise independent random variables in range 0; 1;:::; m − 1. That is we want to generate X0; X2;:::; Xn−1 such that Pr[Xi = α] = 1=m for each α 2 f0; 1; 2;:::; m − 1g Xi and Xj are independent for any i 6= j Chandra and Michael (UIUC) cs473 5 Fall 2019 5 / 45 Relies on the fact that Zp = f0; 1; 2;:::; p − 1g is a field Constructing pairwise independent rvs Suppose we want to create n pairwise independent random variables in range 0; 1;:::; m − 1. That is we want to generate X0; X2;:::; Xn−1 such that Pr[Xi = α] = 1=m for each α 2 f0; 1; 2;:::; m − 1g Xi and Xj are independent for any i 6= j Interesting case: n = m = p where p is a prime number Pick a; b uniformly at random from f0; 1; 2;:::; p − 1g Set Xi = ai + b Only need to store a; b. Can generate Xi from i. Chandra and Michael (UIUC) cs473 5 Fall 2019 5 / 45 Constructing pairwise independent rvs Suppose we want to create n pairwise independent random variables in range 0; 1;:::; m − 1. That is we want to generate X0; X2;:::; Xn−1 such that Pr[Xi = α] = 1=m for each α 2 f0; 1; 2;:::; m − 1g Xi and Xj are independent for any i 6= j Interesting case: n = m = p where p is a prime number Pick a; b uniformly at random from f0; 1; 2;:::; p − 1g Set Xi = ai + b Only need to store a; b. Can generate Xi from i. Relies on the fact that Zp = f0; 1; 2;:::; p − 1g is a field Chandra and Michael (UIUC) cs473 5 Fall 2019 5 / 45 Pairwise independence for general n and m A rough sketch. If n < m we can use a prime p 2 [m; 2m] (one always exists) and use the previous construction based on Zp. n > m is the more difficult case and also relevant. The following is a fundamental theorem on finite fields. Theorem Every finite field F has order pk for some prime p and some integer k ≥ 1. For every prime p and integer k ≥ 1 there is a finite field F of order pk and is unique up to isomorphism. We will assume n and m are powers of 2. From above can assume we have a field F of size n = 2k . Chandra and Michael (UIUC) cs473 6 Fall 2019 6 / 45 Pairwise independence when n, m are powers of 2 We will assume n and m are powers of 2. We have a field F of size n = 2k . Generate n pairwise independent random variables from [n] to [n] by picking random a; b 2 F and setting Xi = ai + b (operations in F). From previous proof X1;:::; Xn are pairwise independent. Now Xi 2 [n]. Truncate Xi to [m] by dropping the most significant log n − log m bits. Resulting variables are still pairwise independent (both n; m being powers of 2 important here). Skipping details on computational aspects of F which are closely tied to the proof of the theorem on fields. Chandra and Michael (UIUC) cs473 7 Fall 2019 7 / 45 Lemma P Suppose X = Xi and X1; X2;:::; Xn are pairwise independent, Pi then Var(X ) = i Var(Xi ). Hence pairwise independence suffices if one relies only on Chebyshev inequality. Pairwise Independence and Chebyshev's Inequality Chebyshev's Inequality Var(X ) For a ≥ 0, Pr[jX − E[X ] j ≥ a] ≤ a2 equivalently for any 1 p t > 0, Pr[jX − E[X ] j ≥ tσX ] ≤ t2 where σX = Var(X ) is the standard deviation of X . Suppose X = X1 + X2 + ::: + Xn. P If X1; X2;:::; Xn are independent then Var(X ) = i Var(Xi ). Chandra and Michael (UIUC) cs473 8 Fall 2019 8 / 45 Pairwise Independence and Chebyshev's Inequality Chebyshev's Inequality Var(X ) For a ≥ 0, Pr[jX − E[X ] j ≥ a] ≤ a2 equivalently for any 1 p t > 0, Pr[jX − E[X ] j ≥ tσX ] ≤ t2 where σX = Var(X ) is the standard deviation of X . Suppose X = X1 + X2 + ::: + Xn. P If X1; X2;:::; Xn are independent then Var(X ) = i Var(Xi ). Lemma P Suppose X = Xi and X1; X2;:::; Xn are pairwise independent, Pi then Var(X ) = i Var(Xi ). Hence pairwise independence suffices if one relies only on Chebyshev inequality. Chandra and Michael (UIUC) cs473 8 Fall 2019 8 / 45 Part II Hash Tables Chandra and Michael (UIUC) cs473 9 Fall 2019 9 / 45 Dictionary Data Structure 1 U: universe of keys with total order: numbers, strings, etc. 2 Data structure to store a subset S ⊆ U 3 Operations: 1 Search/look up: given x 2 U is x 2 S? 2 Insert: given x 62 S add x to S. 3 Delete: given x 2 S delete x from S 4 Static structure: S given in advance or changes very infrequently, main operations are lookups. 5 Dynamic structure: S changes rapidly so inserts and deletes as important as lookups. Can we do everything in O(1) time? Chandra and Michael (UIUC) cs473 10 Fall 2019 10 / 45 Given S ⊆ U. How do we store S and how do we do lookups? Ideal situation: 1 Each element x 2 S hashes to a distinct slot in T . Store x in slot h(x) 2 Lookup: Given y 2 U check if T [h(y)] = y. O(1) time! Collisions unavoidable if jT j < jUj. Hashing and Hash Tables Hash Table data structure: 1 A (hash) table/array T of size m (the table size). 2 A hash function h : U ! f0;:::; m − 1g. 3 Item x 2 U hashes to slot h(x) in T . Chandra and Michael (UIUC) cs473 11 Fall 2019 11 / 45 Hashing and Hash Tables Hash Table data structure: 1 A (hash) table/array T of size m (the table size). 2 A hash function h : U ! f0;:::; m − 1g. 3 Item x 2 U hashes to slot h(x) in T . Given S ⊆ U. How do we store S and how do we do lookups? Ideal situation: 1 Each element x 2 S hashes to a distinct slot in T . Store x in slot h(x) 2 Lookup: Given y 2 U check if T [h(y)] = y. O(1) time! Collisions unavoidable if jT j < jUj. Chandra and Michael (UIUC) cs473 11 Fall 2019 11 / 45 Handling Collisions: Chaining Collision: h(x) = h(y) for some x 6= y. Chaining/Open hashing to handle collisions: 1 For each slot i store all items hashed to slot i in a linked list. T [i] points to the linked list 2 Lookup: to find if y 2 U is in T , check the linked list at T [h(y)]. Time proportion to size of linked list. y f s Does hashing give O(1) time per operation for dictionaries? Chandra and Michael (UIUC) cs473 12 Fall 2019 12 / 45 Hash Functions Parameters: N = jUj (very large), m = jT j, n = jSj Goal: O(1)-time lookup, insertion, deletion. Single hash function If N ≥ m2, then for any hash function h : U! T there exists i < m such that at least N=m ≥ m elements of U get hashed to slot i. Any S containing all of these is a very very bad set for h! Such a bad set may lead to O(m) lookup time! In practice: Dictionary applications: choose a simple hash function and hope that worst-case bad sets do not arise Crypto applications: create \hard" and \complex" function very carefully which makes finding collisions difficult Chandra and Michael (UIUC) cs473 13 Fall 2019 13 / 45 Hashing from a theoretical point of view Consider a family H of hash functions with good properties and choose h randomly from H Guarantees: small # collisions in expectation for any given S.
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