Information Theory

Read e-book online Complexity Theory: Exploring the Limits of Efficient PDF

By Ingo Wegener, R. Pruim

ISBN-10: 3540210458

ISBN-13: 9783540210450

Displays fresh advancements in its emphasis on randomized and approximation algorithms and verbal exchange types All issues are thought of from an algorithmic standpoint stressing the consequences for set of rules layout

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Extra info for Complexity Theory: Exploring the Limits of Efficient Algorithms

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Proof. EP ⊆ ZPP(1/2): If a problem belongs to EP, then there is a randomized algorithm that correctly solves this problem and for every input of length n has an expected runtime that is bounded by a polynomial p(n). 9) says that the probability of a runtime bounded by 2 · p(n) is at least 1/2. So we will stop the algorithm if it has not halted on its own after 2 · p(n) steps. If the algorithm stops on its own (which it does with probability at least 1/2), then it computes a correct result. ”. By definition, this modified algorithm is a ZPP(1/2) algorithm.

Analogously, A, A is (0, 1|0). The combined algorithm (A, A) has three possible results (since (1, 1) is impossible). These results are evaluated as follows: • (1, 0): Since A(x) = 1, x must be in L. ) So we accept x. • (0, 1): Since A(x) = 1, x must be in L. ) So we reject x. ”. The new algorithm is error-free. If x ∈ L, then A(x) = 0 with certainty, and A(x) = 1 with probability at least 1/2, so the new algorithm accepts x with probability at least 1/2. If x ∈ / L, then it follows in an analogous way that the new algorithm rejects with probability at least 1/2.

So we denote by co-RP(ε(n)) the class of languages L for which L ∈ RP(ε(n)). In more detail, this is the class of decision problems that have randomized algorithms with polynomially bounded worst-case runtime that accept every input that should be accepted, and for inputs of length n that should be rejected, have an error-probability bounded by ε(n) < 1. Of course, we can only use algorithms that fail or make errors when the failure- or error-probability is small enough. For time critical applications, we may also require that the worst-case runtime is small.

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Complexity Theory: Exploring the Limits of Efficient Algorithms by Ingo Wegener, R. Pruim


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