Module36 - NP-complete and NP-hard problems•Definition of...

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Unformatted text preview: NP-complete and NP-hard problems•Definition of complexity class NP–Nondeterministic computation–Problems that can be verified•The P = NP Question–Concept of NP-hard and NP-complete problemsTraditional definition of NP•Turing machine model of computation–Simple model where data is on an infinite capacity tape–Only operations are reading char stored in current tape cell, writing a char to current tape cell, moving tape head left or right one square•Deterministic versus nondeterministic computation–Deterministic: At any point in time, next move is determined–Nondeterministic: At any point in time, several next moves are possible•NP: Class of problems that can be solved by a nondeterminstic turing machine in polynomial timeTuring MachinesA Turing machine has a finite-state-control (its program), a two way infinite tape (its memory) and a read-write head (its program counter) 111111HeadTape….….Finite StateControlNondeterministic Running Time•We measure running time by looking at height of computation tree, NOT number of nodes explored•Both computation have same height 4 and thus same running timeDeterministic ComputationNondeterministic ComputationND computation returning yes•If any leaf node returns yes, we consider the input to be a yes input.•If all leaf nodes return no, then we consider the input to be a no input.Yes ResultNo ResultShowing a problem is in NP•Hamiltonian Path–Input: Undirected graph G = (V,E)–Y/N Question: Does G contain a HP?•Nondeterministic polynomial-time solution–Guess a hamiltonian path P (ordering of vertices)•V! possible orderings•For binary tree, V log V height to generate all guesses–Verify guessed ordering is correct–Return yes/no if ordering is actually a HP Illustration123Yes input graph123132213231312321Guess PhaseNondeterministic---------------Verify PhaseDeterministic123No input graph123132213231312321Guess PhaseNondeterministic---------------Verify PhaseDeterministicAlternate definition of NP•Preliminary Definitions–...
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Module36 - NP-complete and NP-hard problems•Definition of...

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