14 - NP Design and Analysis of Algorithms Andrei Bulatov...

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NP Design and Analysis of Algorithms Andrei Bulatov

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Algorithms – NP 14-2 Problems and Algorithms We can encode any combinatorial problem as a binary string The length of a string s is denoted by |s| A decision problem X is the set of strings on which the answer is “yes” An algorithm A for a decision problem receives an input string s and tputs “yes” or “no” outputs “yes” Denote the answer by A(s) The algorithm A solves X if A(s) = “yes” if and only if s X The algorithm A has a polynomial running time if there is a polynomial p such that for every input string s, the algorithm terminates on s in at most O(p(|s|))
Algorithms – NP 14-3 Efficient Certification By a “solution” of a decision problem X we understand a certificate witnessing that an instance is a “yes”-instance We say that an algorithm B is an efficient certifier for a problem X if - B is a polynomial time algorithm that takes two input arguments: instance s and a certificate t - there is polynomial p such that for every string s, we have s S if and only if there exists a string t such that |t| p(|s|) and B(s,t) = yes Denote the answer by A(s) Certifying vs. Solving Certifying and brute force

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14-4 Efficient Certification: Composite COMPOSITES. Given an integer s, is s composite? Certificate. A nontrivial factor t of s. Note that such a certificate exists iff s is composite. Moreover |t| |s|. Certifier. Instance. s = 437,669. Certificate. t = 541 or 809. Conclusion.
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This note was uploaded on 11/11/2009 for the course CS 405/705 taught by Professor Bulatov during the Fall '09 term at Simon Fraser.

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14 - NP Design and Analysis of Algorithms Andrei Bulatov...

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