Polynomial Identity Testing Lemma Bipartite Matching References - PowerPoint PPT Presentation

Polynomial Identity Testing Problems Schwartz-Zippel Polynomial Identity Testing Lemma Bipartite Matching References Anil Maheshwari anil@scs.carleton.ca School of Computer Science Carleton University Canada

String Equality Testing Polynomial Identity Testing Problems Alice-Bob String Testing Problem Schwartz-Zippel Lemma Assume Alice has a binary string A = a 1 a 2 . . . a n and Bob Bipartite Matching has a binary string B = b 1 b 2 . . . b n . What is minimum References amount of communication required to test whether A = B ?

Randomized Algorithm Polynomial Identity Testing n n a i x i and B ( x ) = b i x i , where x ∈ F . � � Define A ( x ) = Problems i =1 i =1 Schwartz-Zippel F is a Field defined with modular arithmetic for a large Lemma Bipartite Matching prime number p . References Algorithm: Pick a random element α ∈ F 1 Alice computes A ( α ) and sends ( α, A ( α )) to Bob 2 Bob computes B ( α ) 3 Bob communicates to Alice True if B ( α ) = A ( α ) , else 4 False

Analysis of Randomized Algorithm Polynomial Identity Testing Case I: A = B : Algorithms reports TRUE as Problems Schwartz-Zippel A ( α ) = B ( α ) no matter what is the value Lemma α ∈ F Bipartite Matching Case II: Assume A � = B . References Can algorithm make an error? Yes, if α is the root of the polynomial ( A − B ) x = 0 . Pr(a random element of F is root of ( A − B ) x ) ≤ n/ | F | Question: How to increase the success probability? Communication Complexity: O (log | F | ) bits.

Multivariate Polynomials Polynomial Identity Testing Determine if the multivariate polynomial Problems Q ( x 1 , x 2 , . . . , x n ) ≡ 0 ? Schwartz-Zippel Lemma Bipartite Matching Example I: References Q ( x 1 , x 2 , x 3 , x 4 ) = ( x 3 1 − x 2 2 )( − x 2 1 − x 4 3 )( x 3 4 − 2 x 1 x 2 ) ≡ 0 Example II: x 1 − x 2 x 2 x 3 − x 1 x 4 − x 1 2 4 − x 4 x 2 2 − x 2 x 2 − x 4 2 x 3 − 7 x 1 2 3 Det ≡ 0 x 3 x 3 x 2 − x 1 x 4 − x 3 1 2 x 3 x 1 − x 2 x 3 1 − x 3 x 4 − 2 x 2 2 3 2

Schwartz–Zippel Lemma Polynomial Identity Testing Problems Schwartz–Zippel Lemma Schwartz-Zippel Lemma Let Q ( x 1 , x 2 , . . . , x n ) �≡ 0 be a multivariate polynomial of Bipartite Matching total degree d , where each x i takes value from a finite References field F . Fix any finite set S ⊆ F and let r 1 , . . . , r n be chosen uniformly at random from S . Then d Pr ( Q ( r 1 , . . . , r n ) = 0) ≤ | S |

Testing Determinants Polynomial Identity Testing x 1 − x 2 x 2 x 3 − x 1 x 4 − x 1 2 4 Problems − x 4 x 2 2 − x 2 x 2 − x 4 2 x 3 − 7 x 1 2 3 Schwartz-Zippel Is Det ≡ 0 ? x 3 x 3 Lemma x 2 − x 1 x 4 − x 3 1 2 Bipartite Matching x 3 x 1 − x 2 x 3 1 − x 3 x 4 − 2 x 2 2 3 2 References Choose a large enough prime number p , and choose random values for x 1 , x 2 , x 3 , x 4 from { 0 , . . . , p − 1 } . Evaluate the determinant. Probability of one sided error ≤ d p , where d is the degree of the polynomial.

Bipartite Matching Polynomial Identity Testing Let G = ( U ∪ V, E ) be a bipartite graph, where Problems | U | = | V | = n . Schwartz-Zippel Lemma M ⊆ E is a perfect matching if Bipartite Matching References | M | = n 1 Edges in M are independent, i.e. vertex disjoint. 2

Adjacency Matrix Polynomial Identity Testing Define n × n matrix A where, Problems Schwartz-Zippel � Lemma x ij , if u i v j ∈ E A ij = Bipartite Matching 0 , otherwise References x 11 x 12 0 x 21 x 22 0 x 31 x 32 x 33 x 11 x 12 x 13 0 x 22 0 0 x 32 0

Edmonds Theorem Polynomial Identity Testing Problems Edmonds Schwartz-Zippel A bipartite graph G has a perfect matching if and only if Lemma Bipartite Matching det ( A ) � = 0 . References

Decision Problem Polynomial Identity Testing Input: Given a bipartite graph G = ( U ∪ V, E ) , where Problems | U | = | V | Schwartz-Zippel Lemma Output: TRUE if G has perfect matching, otherwise Bipartite Matching FALSE References Randomized Algorithm: Choose a large enough prime number p . 1 For each edge u i v j , set x ij to be a random value in 2 { 0 , . . . , p − 1 } uniformly at random. Compute det ( A ) 3 Return TRUE iff det ( A ) � = 0 . 4

Analysis Polynomial Identity Testing Choose a large enough prime number. Problems 1 Schwartz-Zippel For each edge u i v j , set x ij to be a random value in 2 Lemma { 0 , . . . , p − 1 } uniformly at random. Bipartite Matching References Compute det ( A ) 3 Return TRUE iff det ( A ) � = 0 . 4 Case 1: If G has no perfect matching = ⇒ det ( A ) = 0 Case 2: If G has perfect matching = ⇒ det ( A ) � = 0 (Edmonds) Degree of determinant polynomial is ≤ n = | U | Pr(det(A)=0 given that G has a perfect matching) ≤ n/p (Schwartz-Zippel) Choose p ≈ 1000 n , Probability of success ≥ 1 − 1 / 1000

How to find a perfect matching Polynomial Identity Testing Problems Isolation Lemma (MVV87) Schwartz-Zippel Lemma Assume we have set system S on a ground set of n Bipartite Matching elements. Assign weights to each element uniformly and References at random from { 1 , 2 , . . . , 2 n } . The probability that there is a unique minimum weight set in S is ≥ 1 2

Finding a Perfect Matching Polynomial Identity Testing - Let G = ( U ∪ V, E ) and | E | = m . Problems - Assume G has a perfect matching. Schwartz-Zippel Lemma - For each edge e ∈ E , assign a weight in { 1 , . . . , 2 m } Bipartite Matching uniformly at random. References - Let M = Set system consisting of all perfect matchings - Isolation Lemma: ∃ M ∈ M of unique minimum weight with probability ≥ 1 / 2 . New Problem Find (unique) minimum weight perfect matching M in G

Unique MWPM Polynomial Identity Testing Let unique MWPM has a total weight W ≤ 2 m 2 . Problems For each edge e = ( u i v j ) ∈ E with weight w ( e ) , Schwartz-Zippel Lemma set x ij = 2 w ( e ) in det ( A ) . Bipartite Matching References Consider the non-zero terms in the expansion of det ( A ) . Observation: Only one term is 2 W and all other terms are ≥ 2 W +1 = 2 ∗ 2 W .

Unique MWPM (contd.) Polynomial Identity Testing Note: Problems  odd, if k = W Schwartz-Zippel det ( A )  Lemma  = even, if k < W Bipartite Matching 2 k  fractional, if k > W  References Algorithm: Find k : Guess k and check parity of det ( A ) 1 2 k For each edge e = ( uv ) , it is in unique MWPM if and 2 only if MWPM in G \ { u, v } has weight W − w ( e ) . Note: Computation of det( A ), Guess & Check k , and Testing ∀ e ∈ E is part of unique MWPM are parallelizable.

Matching in General Graphs Polynomial Identity Testing Let G = ( V, E ) be a general graph. Problems Schwartz-Zippel Lemma Define Bipartite Matching  + x ij , if v i v j ∈ E and i<j  References  A ij = − x ij , if v i v j ∈ E and i>j  0 , otherwise  Tutte G has a perfect matching if and only if det ( A ) � = 0 .

References Polynomial Identity Testing Mitzenmacher and Upfal, Probability and Computing, Problems 1 Schwartz-Zippel Cambridge. Lemma Motwani and Raghavan, Randomized Algorithm, 2 Bipartite Matching Cambridge. References Mulmuley, Vazirani and Vazirani, Matching is as easy 3 as matrix inversion, Combinatorica 7(1):105-113, 1987. DeMillo and Lipton, A probabilistic remark on 4 algebraic program testing, Information Processing Letters 7(4):193-195, 1978.