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Unformatted text preview: P and NP Definition: P is the class of languages that are decidable in polynomial time on a deterministic Turing machine, P = [ k TIME ( n k ) , for k . Definition: NP = [ k NTIME ( n k ) , for k . Observation: In order to prove that a language is a member of a particular complexity class we simply have to demonstrate than an appropriate algorithm exists. We have seen this in the case of the directed path in a graph. p. 1/1 6 Properties of P and NP Theorem: The complexity class P is closed under complementation. Proof: Any language L P can be decided in deterministic polynomial time. Let M be such a decider for L . To show that P is closed under complementation we show that we can construct a deterministic polynomial time decider M for L , M = "On input w , where w is a string: 1. Run M on w . 2. If M accepts, reject ; if M rejects, accept ." It is easy to see that this machine runs in deterministic polynomial time. a50 p. 2/1 6 Properties of P and NP Theorem: The complexity class NP is closed under the Kleeneclosure. Proof: Any language L in NP is decided by some nondeterministic polynomial time TM. Let M be such a decider for L . To show that NP is closed under the Kleeneclosure we need to show that L NP , where L = { w  w = or w = w 1 w 2 . . . w k for k 1 and each w i L } . We construct a nondeterministic polynomial time TM M that decides L , M = "On input w , where w is a string: 1. If w = , then accept . 2. Nondeterministically split w into the strings w 1 w 2 . . . w k for k 1 . 3. Run M on each string w i . 4. If M accepts all w i s, accept ; otherwise, reject ." By realizing that the number of times the machine M is invoked is bounded by O ( n ) (each  w i  = 1 with n =  w  ) and the fact that M is a nondeterministic polynomial time TM, say O ( n m ) , then the total nondeterministic polynomial runtime is O ( n m +1 ) . Therefore, L NP . a50 p. 3/1 6 Hamiltonian Path A Hamiltonian path in a directed path is a directed path that goes through each node exactly once.A Hamiltonian path in a directed path is a directed path that goes through each node exactly once....
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 Spring '11
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