l15 - 6.896 Sublinear Time Algorithms April 3, 2007 Lecture...

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6.896 Sublinear Time Algorithms April 3, 2007 Lecture 15 Lecturer: Kevin Matulef Scribe: Huy Le Nguyen 1 Defnitions and ProoF Plan Lemma 1 (Szemeredi’s Regularity Lemma). For any ± ,thereex is ts C ± such that any graph G =( V,E ) can be partitioned into sets V 0 ,V 1 ,...,V k k C ± such that 1. | V 0 |≤ ±V 2. | V 1 | = | V 2 | = ... = | V k | 3. All but at most ± ( k 2 ) pairs ( V i j ) are ± -regular Defnition 2 The density between disjoint vertex sets A, B is d ( A, B )= e ( A,B ) | A |·| B | . We also de±ne d ( A, A 2 e ( A,A ) | A | 2 . Defnition 3 AandBare ± -regular if A 0 A, | A 0 | | A | ,B 0 B, | B 0 | | B | , we have | d ( A 0 0 ) d ( A, B ) | . One application of Szemeredi’s Regularity Lemma that we mentioned last time was testing triangle- freeness of graph. The algorithm is very simple, just taking random triples and testing if they form triangles. We used an edge removal argument to prove the correctness of the algorithm. After partitioning the graph into components by Szemeredi’s Regularity Lemma, we cleaned it up by removing internal edges of the components and edges between non ± -regular pairs. Afterwards, since all pairs of components are ± -regular, we can restrict ourselves to testing ±
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l15 - 6.896 Sublinear Time Algorithms April 3, 2007 Lecture...

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