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Unformatted text preview: 6.896 Sublinear Time Algorithms October 7, 2004 Lecture 8 Lecturer: Ronitt Rubinfeld Scribe: Kayi Lee 1 R e v i e w 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, > , 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 > , 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....
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This note was uploaded on 04/02/2010 for the course CS 6.896 taught by Professor Ronittrubinfeld during the Spring '04 term at MIT.
- Spring '04