## Algorithms Sequential & Parallel: A Unified Approach by Russ Miller, Laurence Boxer

By Russ Miller, Laurence Boxer

With multi-core processors exchanging conventional processors and the move to multiprocessor workstations and servers, parallel computing has moved from a distinctiveness quarter to the center of computing device technology. that allows you to offer effective and not pricey options to difficulties, algorithms has to be designed for multiprocessor platforms. Algorithms Sequential and Parallel: A Unified technique 2/E offers a cutting-edge method of an algorithms direction. The e-book considers algorithms, paradigms, and the research of recommendations to severe difficulties for sequential and parallel types of computation in a unified model. this offers practising engineers and scientists, undergraduates, and starting graduate scholars a history in algorithms for sequential and parallel algorithms inside one textual content. necessities contain basics of knowledge constructions, discrete arithmetic, and calculus.

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**Example text**

An optimal algorithm need not be the fastest possible algorithm to give a correct solution to its problem, but it must be within a constant factor of being the fastest possible algorithm to solve the problem. Proving optimality is often difficult and for many problems optimal running times are not known. There are, however, problems for which proof of optimality is fairly easy, some of which will appear in this book. Summary In this chapter, we have introduced fundamental notions and terminology of analysis of algorithms.

Notice that by choice of k0, we must have P(k0 – 1) = true. It follows from step 2 of the Principle of Mathematical Induction that P(k0) = P((k0 – 1) + 1) = true, contrary to the fact that k0 S . Because the contradiction results from the assumption that the principle is false, the proof is established. Induction Examples EXAMPLE n( n + 1) . 2 i =1 Before we give a proof, we show how you might guess the formula to be n Prove that for all positive integers n, ¨i = n proven if it weren’t already given to you.

Why does this work? Suppose we have accomplished the two steps given above. Roughly speaking (we’ll give a mathematically stronger argument next), we know from step 1 that P(1) is true, and thus by step 2 that P(1 + 1) = P(2) is true, P(2 + 1) = P(3) is true, P(3 + 1) = P(4) is true, and so forth. That is, step 2 allows us to induce the truth of P(n) for every positive integer n from the truth of P(1). The assumption in step 2 that P(k) = true is called the inductive hypothesis, because it is typically used to induce the conclusion that the successor statement P(k + 1) is true.