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Unformatted text preview: 1 Intractable Problems TimeBounded Turing Machines Classes P and NP PolynomialTime Reductions 2 TimeBounded TMs A Turing machine that, given an input of length n, always halts within T(n) moves is said to be T(n)time bounded . The TM can be multitape. Sometimes, it can be nondeterministic. The deterministic, multitape case corresponds roughly to an O(T(n)) runningtime algorithm. 3 The class P If a DTM M is T(n)time bounded for some polynomial T(n), then we say M is polynomialtime ( polytime ) bounded. And L(M) is said to be in the class P . Important point : when we talk of P , it doesnt matter whether we mean by a computer or by a TM (next slide). 4 Polynomial Equivalence of Computers and TMs A multitape TM can simulate a computer that runs for time O(T(n)) in at most O(T 2 (n)) of its own steps. If T(n) is a polynomial, so is T 2 (n). 5 Examples of Problems in P Is w in L(G), for a given CFG G? Input = w. Use CYK algorithm, which is O(n 3 ). Is there a path from node x to node y in graph G? Input = x, y, and G. Use Dijkstras algorithm, which is O(n log n) on a graph of n nodes and arcs. 6 Running Times Between Polynomials You might worry that something like O(n log n) is not a polynomial. However, to be in P , a problem only needs an algorithm that runs in time less than some polynomial. Surely O(n log n) is less than the polynomial O(n 2 ). 7 A Tricky Case : Knapsack The Knapsack Problem is: given positive integers i 1 , i 2 ,, i n , can we divide them into two sets with equal sums?...
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 Spring '08
 Motwani,R

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