For many applications, a randomized algorithm is either the simplest or the fastest algorithm available, and sometimes both. This book introduces the basic concepts in the design and analysis of randomized algorithms. The first part of the text presents basic tools such as probability theory and probabilistic analysis that are frequently used in algorithmic applications. Algorithmic examples are also given to illustrate the use of each tool in a concrete setting. In the second part of the book, each chapter focuses on an important area to which randomized algorithms can be applied, providing a comprehensive and representative selection of the algorithms that might be used in each of these areas. Although written primarily as a text for advanced undergraduates and graduate students, this book should also prove invaluable as a reference for professionals and researchers.

Randomized Algorithms

Optimal aggregation algorithms for middleware

If the Internet is the next great subject for Theoretical Computer Science to model and illuminate mathematically, then Game Theory, and Mathematical Economics more generally, are likely to prove useful tools. In this talk I survey some opportunities and challenges in this important frontier.

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Algorithms, games, and the internet

There has been a great deal of interest recently in the relative power of on-line and off-line algorithms. An on-line algorithm receives a sequence of requests and must respond to each request as soon as it is receiveD. An off-line algorithm may wait until all requests have been received before determining its responses. One approach to evaluating an on-line algorithm is to compare its performance with that of the best possible off-line algorithm for the same problem. Thus, given a measure of "profit", the performance of an on-line algorithm can be measured by the worst-case ratio of its profit to that of the optimal off-line algorithm. This general approach has been applied in a number of contexts, including data structures [SITa], bin packing [CoGaJo], graph coloring [GyLe] and the k-server problem [MaMcSI]. Here we apply it to bipartite matching and show that a simple randomized on-line algorithm achieves the best possible performance.

/pdf/an-optimal-algorithm-for-on-line-bipartite-matching-452vu1mkc2.pdf

An optimal algorithm for on-line bipartite matching

1. Introduction The study of expected running time of algoritruns is an interesting subject from both a theoretical and a practical point of view. Basically there exist two approaches to this study. In the first approach (we shall call it the distributional approach), some "natural" distribution is assumed for the input of a problem, and one looks for fast algorithms under this assumption (see Knuth [8J). For example, in sorting n numbers, it is usually assumed that all n! initial orderings of the numbers are equally likely. A common criticism of this approach is that distributions vary a great deal in real life situations; fu.rthermore, very often the true distribution of the input is simply not known. An alternative approach which attempts to overcome this shortcoming by allowing stochastic moves in the computation has recently been proposed. This is the randomized approach made popular by Habin [lOJ(also see Gill[3J, Solovay and Strassen [13J), although the concept was familiar to statisticians (for exa'1lple, see Luce and Raiffa [9J). Note that by allowing stochastic moves in an algorithm, the input is effectively being randomized. We shall refer to such an algoritlvn as a randomized algorithm. These two approaches lead naturally to two different definitions of intrinsic complexity of a problem, which we term the distributional complexity and the randomized complexity, respectively. (Precise definitions and examples will be given in Sections 2 and 3.) To solidify the ideas, we look at familiar combinatorial problems that can be modeled by decision trees. In particular, we consider (a) the testing of an arbitrary graph property from an adjacency matrix (Section 2), and (b) partial order problems on n We will show that for these two classes of problems, the two complexity measures always agree by virtue of a famous theorem, the Minimax Theorem of Von Neumann [14J. The connection between the two approaches lends itself to applications. With two different views (and in a sense complementary to each other) on the complexity of a problem, it is frequently easier to derive upper and lower bounds. For example, using adjacency matrix representation for a graph, it can be shown that no randomized algorithm can determine 2 the existence of a perfect matching in less than O(n) probes. Such lower bounds to the randomized approach were lacking previously. As another example of application , we can prove that for the partial order problems in (b), assuming uniform …

Andrew Chi-Chin Yao

Papers

Probabilistic computations: Toward a unified measure of complexity