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By Ingo Wegener, R. Pruim

Displays fresh advancements in its emphasis on randomized and approximation algorithms and communique versions All issues are thought of from an algorithmic standpoint stressing the results for set of rules layout

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

Example text

Modern cryptography (see Stinson (1995)) has a tight connection to number theoretic problems in which very large numbers are used. Here we must take note that the binary representation of an input n has a length of only log(n + 1) . Already in gradeschool, most of us learned an algorithm for adding fractions that required us to compute common denominators, and for this we factored the denominators into prime factors. This is the problem of factoring (Fact). Often it suffices to check whether a number is prime (primality testing).

If we take the word ‘verification’ seriously, then the first type of error must not be allowed. 4. We will denote by RP(ε(n)) (random polynomial time) the class of decision problems for which there is an algorithm with polynomially bounded worst-case computation time and the following acceptance behavior: Every input that should be rejected, is rejected; and for every input of length n that should be accepted, the probability of erroneously rejecting the input is bounded by ε(n) < 1. This type of error is known as one-sided error, in contrast to the two-sided error that is allowed in BPP(ε(n)) algorithms.

There the expected value was taken with respect to a probability distribution on the inputs of length n, which is unknown in practice. Here the expected value is taken with respect to the random bits, the quality of which we can control. Purists could complain that in our example of QuickSort, random bits only allow for the random choice from among n objects when n is a power of two (n = 2k ). But this is not a problem. We can work in phases during which we read log n random bits. These log n random bits can be interpreted as a random number z ∈ {0, .

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