Schwartz-Zippel Algorithm and Bipartite Perfect Matching: Lecture 14 - Prof. Andrew Mcgreg, Exams of Computer Science

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CMPSCI 711: “Really Advanced Algorithms”

Lecture 14 – Finger Printing

Andrew McGregor

Last Compiled: March 30, 2009

Schwartz-Zippel

String Equality and Some Communication Complexity

Pattern Matching

Readings

Schwartz-Zippel

Problem

Given three n variable polynomials P 1 , P 2 , P 3 over the field F. Can you test if

P 1 (x 1 ,... , xn) × P 2 (x 1 ,... , xn) = P 3 (x 1 ,... , xn)

faster than multiplying the polynomials? Equivalently, is

Q(x 1 ,... , xn) = P 1 (x 1 ,... , xn) × P 2 (x 1 ,... , xn) − P 3 (x 1 ,... , xn)

zero for all x 1 ,... , xn?

Schwartz-Zippel

Problem

Given three n variable polynomials P 1 , P 2 , P 3 over the field F. Can you test if

P 1 (x 1 ,... , xn) × P 2 (x 1 ,... , xn) = P 3 (x 1 ,... , xn)

faster than multiplying the polynomials? Equivalently, is

Q(x 1 ,... , xn) = P 1 (x 1 ,... , xn) × P 2 (x 1 ,... , xn) − P 3 (x 1 ,... , xn)

zero for all x 1 ,... , xn?

Theorem (Schwartz-Zippel)

Let Q(x 1 ,... , xn) be a non-zero multivariate polynomial F of total degree d. Fix any finite set S ⊂ F and let r 1 ,... , rn be chosen independently and uniformly at random from S. Then,

P [Q(r 1 ,... , rn) = 0] ≤ d/|S|

Schwartz-Zippel Proof

I (^) Induction on n: For n = 1, because Q has at most d roots, P [Q(r 1 ) = 0] ≤ d/|S| I (^) For induction step define Qi for 0 ≤ i ≤ k:

Q(x 1 ,... , xn) =

∑^ k

i=

xi^ Qi (x 2 ,... , xn)

where k is maximum such that Qk (x 2 ,... , xn) 6 ≡ 0

Schwartz-Zippel Proof

I (^) Induction on n: For n = 1, because Q has at most d roots, P [Q(r 1 ) = 0] ≤ d/|S| I (^) For induction step define Qi for 0 ≤ i ≤ k:

Q(x 1 ,... , xn) =

∑^ k

i=

xi^ Qi (x 2 ,... , xn)

where k is maximum such that Qk (x 2 ,... , xn) 6 ≡ 0 I (^) Since total degree of Qk is at most d − k, P [Qk (r 2 ,... , rn) = 0] ≤ (d − k)/|S|

Schwartz-Zippel Proof

I (^) Induction on n: For n = 1, because Q has at most d roots, P [Q(r 1 ) = 0] ≤ d/|S| I (^) For induction step define Qi for 0 ≤ i ≤ k:

Q(x 1 ,... , xn) =

∑^ k

i=

xi^ Qi (x 2 ,... , xn)

where k is maximum such that Qk (x 2 ,... , xn) 6 ≡ 0 I (^) Since total degree of Qk is at most d − k, P [Qk (r 2 ,... , rn) = 0] ≤ (d − k)/|S| I (^) Consider q(x) = Q(x, r 2 ,... , rn), P [q(r 1 ) = 0|Qk (r 2 ,... , rn) 6 = 0] ≤ k/|S| I (^) Putting together gives P [Q(r 1 ,... , rn) = 0] at most P [Qk (r 2 ,... , rn) = 0]+P [q(r 1 ) = 0|Qk (r 2 ,... , rn) 6 = 0] ≤ d/|S|

Bipartite Perfect Matching

Definition

Let G = (U, V , E ) be a bipartite graph on U = {u 1 ,... , un} and V = {v 1 ,... , vn}. M ⊂ E is a matching if each vertex occurs at most once in M. If |M| = n then we say M is a perfect matching.

Bipartite Perfect Matching

Definition

Let G = (U, V , E ) be a bipartite graph on U = {u 1 ,... , un} and V = {v 1 ,... , vn}. M ⊂ E is a matching if each vertex occurs at most once in M. If |M| = n then we say M is a perfect matching.

Theorem (Edmonds’ Theorem)

Given G , let A be n × n matrix where

Ai,j =

xij if (ui , vj ) ∈ E 0 if (ui , vj ) ∈ E

Then det(A) is multivariate polynomial with maximum degree n. det(A) ≡ 0 iff G has a perfect matching. Schwartz-Zippel result gives randomized method for seeing if G has perfect matching. But it’s actually not that interesting...

Outline

Schwartz-Zippel

String Equality and Some Communication Complexity

Pattern Matching

Readings

Verifying Equality of Strings

Problem

Suppose Alice has binary string (a 1 ,... , an) and Bob has binary string (b 1 ,... , bn). How many bits to this need to communicate to conclude (with high probability) that the strings are equal?

Protocol

I (^) Alice and Bob defines a =

i∈[n] ai^2 n−i (^) and b =

i∈[n] bi^2 n−i

Verifying Equality of Strings

Problem

Suppose Alice has binary string (a 1 ,... , an) and Bob has binary string (b 1 ,... , bn). How many bits to this need to communicate to conclude (with high probability) that the strings are equal?

Protocol

I (^) Alice and Bob defines a =

i∈[n] ai^2 n−i (^) and b =

i∈[n] bi^2 n−i I (^) Alice randomly picks a prime p ≤ τ = tn log tn

Verifying Equality of Strings

Problem

Suppose Alice has binary string (a 1 ,... , an) and Bob has binary string (b 1 ,... , bn). How many bits to this need to communicate to conclude (with high probability) that the strings are equal?

Protocol

I (^) Alice and Bob defines a =

i∈[n] ai^2 n−i (^) and b =

i∈[n] bi^2 n−i I (^) Alice randomly picks a prime p ≤ τ = tn log tn I (^) Alice transmits Fp (a) = a mod p and p to Bob I (^) Bob computes Fp (b): Returns “equal” iff Fp (a) = Fp (b)

Verifying Equality of Strings

Problem

Suppose Alice has binary string (a 1 ,... , an) and Bob has binary string (b 1 ,... , bn). How many bits to this need to communicate to conclude (with high probability) that the strings are equal?

Protocol

I (^) Alice and Bob defines a =

i∈[n] ai^2 n−i (^) and b =

i∈[n] bi^2 n−i I (^) Alice randomly picks a prime p ≤ τ = tn log tn I (^) Alice transmits Fp (a) = a mod p and p to Bob I (^) Bob computes Fp (b): Returns “equal” iff Fp (a) = Fp (b)

Theorem

Protocol uses O(log(tn)) bits of communication and is correct with probability 1 − O(1/t).