# l12 - 6.896 Sublinear Time Algorithms Lecture 12 Lecturer...

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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, we’ve 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 diﬃcult, 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 Lecture 12 Lecturer...

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