By Christian Ronse

ISBN-10: 3540133305

ISBN-13: 9783540133308

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**Example text**

Primes, the problem of deciding whether or not a positive integer n is prime, was our ﬁrst example of an algorithmic problem on page 12. With this colorful bouquet of central and practical algorithmic problems we can discuss most complexity theoretical questions. 3 How Is the Computation Time of an Algorithm Measured? A ﬁrst attempt at a deﬁnition of the complexity of an algorithmic problem might look like the following: The complexity of an algorithmic problem is the amount of computation time required by an optimal algorithm.

Since s ≥ 1 − ε(n), this bound is largest when s = 1 − ε(n). , a polynomial, we obtain an errorprobability that is bounded by 2−q(n) . For most optimization problems we can determine in (deterministic) polynomial time if two results have the same quality. Since the value of the optimal solution is unique, we can in this case reduce the error-probability in an analogous manner. 6. Let p(n) and q(n) be polynomials. If we restrict our attention to the class of problems with a unique solution or to optimization problems for which the value of a solution can be computed in polynomial time, then BPP(1/2 − 1/p(n)) = BPP(2−q(n) ) .

EP = ZPP(1/2). 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.

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