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Unformatted text preview: 6.896 Sublinear Time Algorithms April 5, 2007 Lecture 16 Lecturer: Ronitt Rubinfeld Scribe: Adriana Lopez Recall from last lecture: Definition 1 The density between two disjoint non-empty sets A, B ⊆ V is given by the formula d ( A, B ) = e ( A,B ) | A || B | , where e ( A, B ) is the number of edges between A and B . Definition 2 We say A, B ⊆ V are γ-regular if ∀ A ⊆ A, B ⊆ B such that | A | ≥ γ | A | , | B | ≥ γ | B | , the following inequality holds: | d ( A , B ) − d ( A, B ) | < γ . Lemma 3 (Szemeredi’s Regularity Lemma) For all m and for any > , there exists T = T ( m, ) such that if G = ( V, E ) with | V | > T , and if A is an equipartition of V into m sets, then there exists an equipartition B , a refinement of A , with k sets such that m ≤ k ≤ T and at most ( k 2 ) set pairs are not-regular. This lemma says that any large enough graph can in some sense have a constant-size description that makes it look random with respect to the densities of set pairs. The constant T , however, is very large: T ≈ 2 2 2 2 ··· 1 5 times We will use the regularity lemma to prove the existence of a constant-time tester for triangle freeness, and therefore the running time of the tester will just as large (constant, but large). 1 Testing Triangle Freeness Recall the following lemma from last lecture. We will use this lemma, along with theorem 5, to create a constant-time tester for triangle freeness.a constant-time tester for triangle freeness....
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This note was uploaded on 04/02/2010 for the course CS 6.896 taught by Professor Ronittrubinfeld during the Fall '04 term at MIT.
- Fall '04