# l17 - 6.896 Sublinear Time Algorithms Lecture 17 Lecturer...

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Unformatted text preview: 6.896 Sublinear Time Algorithms April 10, 2007 Lecture 17 Lecturer: Ronitt Rubinfeld Scribe: Jacob Scott 1 Alon’s Theorem (adjacency matrix model) Theorem 1 Let H be a fixed graph with | H | = h , and P H be the property of not having H as a subgraph. Then, 1. If H is bipartite: (a) ∃ a 2-sided tester for P ( H ) with O (1 / ) queries (b) ∃ a 1-sided tester for P ( H ) with O ( h 2 (1 / ) h 2 / 4 ) queries 2. If H is not bipartite, ∃ c such that any 1 − sided error tester needs ≥ ( c ) c log(1 / ) We will prove (2) for triangles, that is, that testing triangle-freeness with 1-sided error requires queries superpolynomial in . 2 Structure of the Algorithm (Goldreich Trevisan) Theorem 2 Let P be any graph property (in the adjacency matrix model). Suppose T is a tester for P with query complexity q ( n, ) . Then there is a tester T for P such that T has the following form: 1. Select a random subset of 2 q ( n, ) nodes 2. Query all pairs in subset 3. Flip coins and make a decision3....
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l17 - 6.896 Sublinear Time Algorithms Lecture 17 Lecturer...

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