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Unformatted text preview: Harvard CS 121 and CSCI E207 Lecture 20: NP Harry Lewis November 19, 2009 • Reading: Sipser § 7.3. Harvard CS 121 & CSCI E27 November 19, 2009 Another way of looking at P • Multiplicative increases in time or computing power yield multiplicative increases in the size of problems that can be solved • If L is in P, then there is a constant factor k such that • If you can solve problems of size s within a given amount of time • and you are given a computer that runs twice as fast, then • you can solve problems of size k · s on the new machine in the same amount of time. • E.g. if L is decidable in O ( n d ) time, then with twice as much time you can solve problems 2 1 d as large 1 Harvard CS 121 & CSCI E27 November 19, 2009 Exponential time • E = ∪ c> TIME ( c n ) • For problems in E, a multiplicative increase in computing power yields only an additive increase in the size of problems that can be solved. • If L is in E, then there is a constant k such that • If you can solve problems of size s within a given amount of time • and you are given a computer that runs twice as fast, then • you can solve problems only of size k + s on the new machine using the same amount of time. 2 Harvard CS 121 & CSCI E27 November 19, 2009 “Nondeterministic Time” • We say that a nondeterministic TM M decides a language L iff for every w ∈ Σ * , 1. Every computation by M on input w halts (in state q accept or state q reject ); 2. w ∈ L iff there exists at least one accepting computation by M on w . 3. w / ∈ L iff every computation by M on w rejects. • M decides L in nondeterministic time t ( · ) iff for every w , every computation by M on w is of length at most t (  w  ) 3 Harvard CS 121 & CSCI E27 November 19, 2009 More on Nondeterministic Time 1. Linear speedup holds. 2. “Polynomial equivalence” holds among nondeterministic models e.g. L decided in time T by a nondeterministic multitape TM ⇒ L decided in time O ( T 2 ) by a nondeterministic 1tape TM Definition: NTIME ( t ( n )) = { L : L is decided in time t ( n ) by some nondet. multitape TM } NP = S polynomial p NTIME ( p ) = S k ≥ NTIME ( n k ) . 4 Harvard CS 121 & CSCI E27 November 19, 2009 P vs. NP • Clearly P ⊆ NP. But there are problems in NP that are not obviously in P ( 6 = “obviously not”) • TSP = T RAVELLING S ALESMAN P ROBLEM . • Let m > be the number of cities , • D : { 1 ,...,m } 2 → N give the distance D ( i,j ) between city i and city j , and • B be a distance bound Then TSP = {h m,D,B i : ∃ tour of all cities of length ≤ B } ....
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 Fall '09
 Computational complexity theory, Prime number, decision problem, Cook–Levin theorem, P versus NP problem

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