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Unformatted text preview: 6.896 Sublinear Time Algorithms April 12, 2007 Lecture 18 Lecturer: Ronitt Rubinfeld Scribe: Ivaylo Riskov 1 R e c a p Theorem 1 1sided error testing for trianglefree property requires Ω( ( c ) c log c ) . Theorem 2 Let P be any graph property in the adjacency matrix model. Let T be a tester for P with query complexity q ( n, ) . Then there is a tester T for P , such that T has form: 1. Selects random subset of 2 q ( n, ) nodes. 2. Queries all pairs. 3. Outputs decision. If T has 1sided error, so does T . Theorem 3 ∀ m ∃ X ⊂ M = { 1 , . . . , m } of size  X  ≥ m e 10 √ log m with no nontrivial solutions to x 1 + x 2 = x 3 . 2 Construction of a graph that is hard to test In this section we present two attempts to construct a graph that is far from trianglefree, but is also hard to test. 2.1 First attempt Consider the following sets of vertices V 1 = { 1 . . . m } , V 2 = { 1 . . . 2 m } , V 3 = { 1 . . . 3 m } . The edges are defined as follows: • For every j ∈ V 1 put an edge ( j, j + x ) from V 1 to V 2 , ∀ x ∈ X . i.e. this is the set of edges { ( j, j + x 1 ) , ( j, j + x 2 ) , . . . } . • For every l ∈ V 2 put an edge ( l, l + x ) from V 1 to V 3 , ∀ x ∈ X . • For every k ∈ V 2 put an edge ( k, k + 2 x ) from V 2 to V 3 , ∀ x ∈ X . The number of nodes is 6 m and the number of vertices is Θ( m.  X  ) = Θ m 2 e 10 √ log m ....
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 Fall '04
 RonittRubinfeld
 Algorithms, Graph Theory, trianglefree

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