16 - PSPACE Design and Analysis of Algorithms Andrei...

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PSPACE Design and Analysis of Algorithms Andrei Bulatov
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Algorithms – PSPACE 16-2 Games Geography . Amy names capital city c of country she is in. Bob names a capital city c' that starts with the letter on which c ends. Amy and Bob repeat this game until one player is unable to continue. Does Alice have a forced win? Example Budapest Tokyo Ottawa Ankara Amsterdam Moscow Washington Nairobi Is Geography in NP?
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Algorithms – PSPACE 16-3 More Games Geography on graphs. Given a directed graph G = (V, E) and a start node s, two players alternate turns by following, if possible, an edge out of the current node to an unvisited node. Can first player guarantee to make the last legal move? Strategies: Is my opponent going left or right?
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Algorithms – PSPACE 16-4 PSPACE P. Decision problems solvable in polynomial time. PSPACE. Decision problems solvable in polynomial space. Observation. P PSPACE. poly-time algorithm can consume only polynomial space
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Algorithms – PSPACE 16-5 3-SAT in PSPACE Proof inary counter: Count from 0 to in binary. Theorem 3-SAT PSPACE n Binary counter: Algorithm: Use n bit odometer. To solve 3-SAT: Enumerate all possible truth assignments using counter. Check each assignment to see if it satisfies all clauses QED n 2 1 2 -
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Algorithms – PSPACE 16-6 NP in PSPACE Proof onsider arbitrary problem X in NP. Theorem NP PSPACE Consider arbitrary problem Since X 3-SAT, there exists algorithm that solves X in poly-time plus polynomial number of calls to 3-SAT black box. Can implement black box in poly-space QED
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Algorithms – PSPACE 16-7 Quantified Satisfiability QSAT Instance A Boolean CNF formula Φ (x 1 , …, x n ) Objective the following propositional formula true? Is the following propositional formula true? 5 x 1 2200 x 2 5 x 3 2200 x 4 2200 x n-1 5 x n Φ (x 1 , …, x n ) Intuition: Amy picks truth value for x 1 , then Bob for x 2 , then Amy for x 3 , and so on. Can Amy satisfy Φ no matter what Bob does? assume n is odd
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16-8 Quantified Satisfiability Example : Yes. Amy sets x 1 true; Bob sets x 2 ; Amy sets x 3 to be same as x 2 . xample ( x 1 x 2 ) ( x 2 x 3 ) ( x 1 x 2 x 3 ) Example : No. If Amy sets x 1 false; Bob sets x 2 false; Amy loses; No. if Amy sets x 1 true; Bob sets x 2 true; Amy loses. ( x
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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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16 - PSPACE Design and Analysis of Algorithms Andrei...

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