# l8 - 6.896 Sublinear Time Algorithms October 7 2004 Lecture...

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6.896 Sublinear Time Algorithms October 7, 2004 Lecture 8 Lecturer: Ronitt Rubinfeld Scribe: Kayi Lee 1 Review In the previous lecture, we introduced the Szemer´ edi’s Regularity Lemma, which states that all graphs can be approximated by random-looking graphs in a certain sense. Definition 1 A pair of disjoint vertex sets ( A, B ) is γ - regular if, for all A A, B B such that | A | ≥ γ | A | and | B | ≥ γ | B | , we have | d ( A, B ) d ( A , B ) | < γ , where d ( A, B ) = e ( A,B ) | A || B | is the density between the sets A and B . Note that if ( A, B ) is γ -regular, ( A, B ) is also γ -regular for all γ γ 1 by definition. Lemma 2 (Szemer´ edi’s Regularity Lemma) For all m, > 0 , there exists T such that if G = ( V, E ) with | V | > T and A is a equipartition of V into m sets, then there exists a equipartition B , a refinement of A , with k sets such that: 1. m k T ; and 2. there are fewer than k 2 pairs of set partitions that are not -regular We also proved the following lemma, which states that a triplet of pairwise regular vertex sets behaves like a random graph with respect to the number of distinct triangles in the graph. Lemma 3 (Komlos and Simonovits) For all η > 0 , there exists γ Δ and δ Δ such that if A, B, C are disjoint subsets of V and each pair is γ Δ -regular with density at least η , then G contains at least δ Δ | A || B || C | distinct triangles with vertex from each of A, B and C . Note that the constants T, γ Δ and δ Δ in the above two lemmas depend on the choices of m, and η , but not the size of the graph. In the rest of the note, we will denote them as as T ( m, ) , γ Δ ( η ) and δ Δ ( η ) to remind readers of the parameters these constants depend on. 1

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2 A 1-sided Triangle-Free Property Tester We have described a 1-sided triangle-free property tester that is based on the following theorem by Alon (see also [Alon Fischer Krivilevich Szegedy]): Theorem 4 For all , there exists δ such that any n -vertex graph G = ( V, E ) that is
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