This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 nonempty 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 γregular 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 notregular. This lemma says that any large enough graph can in some sense have a constantsize 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 constanttime 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 constanttime tester for triangle freeness.a constanttime tester for triangle freeness....
View
Full
Document
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
 RonittRubinfeld
 Algorithms

Click to edit the document details