15-time-complexity

# 15-time-complexity - Nondeterministic Time Complexity Let N...

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Nondeterministic Time Complexity Definition: Let N be a nondeterministic Turing machine that is a decider. The running time of N is the function f : N N , where f ( n ) is the maximum number of steps that N uses on any branch of its computation on any input of length n . –p

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Some Theorems Theorem: Let t ( n ) be a function, where t ( n ) n , then every t ( n ) time multitape Turing machine has an equivalent O ( t 2 ( n )) time single tape Turing machine. Proof Sketch: It is possible to show that simulating each of the t ( n ) computation steps of the multitape machine on a single tape machine takes at most O ( t ( n )) steps. Therefore, to simulate the complete multitape computation on a single tape machine will take t ( n ) × O ( t ( n )) = O ( t 2 ( n )) steps. Observation: Moving a computation from a multi-tape machine to a single-tape machine incurs an polynomial runtime penalty. –p
Some Theorems Theorem: Let t ( n ) be a function where t ( n ) 0 , then every t ( n ) time nondeterministic single-tape Turing machine has an equivalent 2 O ( t ( n )) time deterministic single-tape Turing machine. Proof Sketch: Recall that simulating a nondeterministic Turing machine with a deterministic Turing machine can be viewed as searching the tree of nondeterministic computations for accepting states. Since the nondeterministic TM is a O ( t ( n )) time machine, the path from the root to a leaf node is bounded by

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15-time-complexity - Nondeterministic Time Complexity Let N...

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