l12 - 6.896 Sublinear Time Algorithms March 15, 2007...

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Unformatted text preview: 6.896 Sublinear Time Algorithms March 15, 2007 Lecture 12 Lecturer: Ronitt Rubinfeld Scribe: Jeremy Hurwitz 1 Intro to Property Testers 1.1 Informal Overview So far, weve dealt with two main types of approximations - multiplicative and additive. Such approx- imations work well for optimization problems and problems whose output is over an interval of reals, such as the distance between two distributions or the entropy of a distribution. Unfortunately, such a definition of approximately fails for decision problems. For example, we may want to ask Is G connected? or Is G bipartite? How, then, do we approximate a yes/no answer? Instead of approximating the answer, we will approximate the input. Is G connected? becomes Is G almost a connected graph? Such questions are often dicult, but in some cases we can answer such questions in sublinear time. A property tester is an algorithm that answers this question. The tester outputs yes with high probability if the input has the desired property and answers no with high probability if the input is-far from having the property. 1.2 Formal Definition Given a property P and a domain D , let P = { x D | x has property P } . Definition 1 (Decision Algorithm) A decision algorithm A is defined by if x P , A ( x ) = pass if x 6 P , A ( x ) = fail Definition 2 (-far from P ) Given a metric d ( x, y ) on D , let d ( x, P ) = min...
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l12 - 6.896 Sublinear Time Algorithms March 15, 2007...

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