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# l13 - 6.896 Sublinear Time Algorithms Lecture 13 Lecturer...

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6.896 Sublinear Time Algorithms March 20, 2007 Lecture 13 Lecturer: Ronitt Rubinfeld Scribe: Huy Ngoc Nguyen 1 Bipartiteness testing on dense graph Definition 1 G is ” far ” from bipartiteness if must remove at least n 2 edges to make it bipartite. Definition 2 (equivalent to definition 1) partitions ( V 1 , V 2 ) of V there are at least n 2 edges ( w, v ) s.t either u, v V 1 or u, v V 2 (in this case ( u, v ) is a violating pair ). Algorithm (1): : TestBipartite( A , G ) ( A - adjacency matrix for G ) choose sample S uniformly s.t | S | = Θ( 1 2 log 1 ) 1 query A ( u, v ) u, v S . 2 FAIL if subgraph not bipartite else PASS. 3 Note: total number of queries : O ( 1 ε 4 log 2 1 ε ) Theorem 1 The algorithm is a property tester for bipartiteness. More precisely, (1) if G is bipartite, TestBipartite PASSES. (2) if G is ε far from bipartiteness, Pr [ TestBipartite FAILS ] 3 4 . Proof Consider the following algorithm, Algorithm (2) : Choose U uniformly, s.t | U | = O ( 1 ε log 1 ε ) 1 Choose a bunch of pairs ( u i , v i ) W ( | W | = m = O ( 1 ε 2 log 1 ε ) ) 2 Query ( u, v ) u, v

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l13 - 6.896 Sublinear Time Algorithms Lecture 13 Lecturer...

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