# Design and Analysis of Randomized Algorithms: Introduction to Design Paradigms (Texts in Theoretical Computer Science)

## Juraj Hromkovič

Language: English

Pages: 280

ISBN: B010DQ32QE

Format: PDF / Kindle (mobi) / ePub

Randomness is a powerful phenomenon that can be harnessed to solve various problems in all areas of computer science. Randomized algorithms are often more efficient, simpler and, surprisingly, also more reliable than their deterministic counterparts. Computing tasks exist that require billions of years of computer work when solved using the fastest known deterministic algorithms, but they can be solved using randomized algorithms in a few minutes with negligible error probabilities.

Introducing the fascinating world of randomness, this book systematically teaches the main algorithm design paradigms – foiling an adversary, abundance of witnesses, fingerprinting, amplification, and random sampling, etc. – while also providing a deep insight into the nature of success in randomization. Taking sufficient time to present motivations and to develop the reader's intuition, while being rigorous throughout, this text is a very effective and efficient introduction to this exciting field.

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later one after each other during the computation. But this representation of the second model by the first model is not uniform31 and so we do not see any possibility of expressing any randomized algorithm described by the second model in terms of the first model as a probability distribution over a finite set of deterministic algorithms. 2.4 Classification of Randomized Algorithms The aim of this section is to introduce the fundamental and generally accepted classification of randomized

saves so much essential information about a that using h(a) one can distinguish a from almost all other keys in S for most sets S (or that using h(a) one can distinguish a from most other keys in the universe). It is important to observe that one cannot strengthen the requirement (ii) for the choice of h by requiring a uniform distribution of S in T for every set S ⊆ U . For every hash function h : U → T and every slot i ∈ {0, 1, . . . , m−1}, there exists the set Uh,i = {a ∈ U | h(a) = i}, all

machine Mij . (iii) The execution of a task on its machine costs exactly one time unit. 122 3 Foiling the Adversary If one has m pairwise diﬀerent machines and d tasks, we speak of the so-called Unit (m, d) Job problem. For every positive integer d, we denote by Unit (d) Job the problem ∞ Unit (m, d) Job. m=1 A feasible solution of an instance of Unit (m, d) Job corresponds to a distribution of the tasks to the machines in discrete time. More precisely, a machine and a time unit are

3.4(a) as hatched squares. Figure 3.4(a) shows Grid9 (α, β) for α = (1, 2, 3, 4, 5, 6, 7, 8, 9) and β = (1, 3, 2, 6, 5, 4, 8, 7, 9). The motivation to introduce the term obstacle (collision) is as follows. Assume that the executions of the first l − 1 tasks of job α and of the first k − 1 tasks of job β have finished. Now, if il ̸= jk , one can continue by executing the l-th task of α on the il -th machine Mil and the k-th task of β on the jk -th machine Mjk in parallel. But, if il = jk , then

mentioned deterministic algorithm can compute an optimal cut of G/F in time O (l(n))3 . Altogether, the complexity of DETRAN(l) is in O n2 + (l(n))3 . Next we analyze the success probability of DETRAN(l). As in the proof of Theorem 5.2.1, let Cmin be a minimal cut of G. We prove a lower bound on the success probability of DETRAN(l) by proving a lower bound on the probability of having Cmin in the multigraph G/F after executing step 1. The corresponding event is n−l(n) Eventi . i=1 Applying