This small book grew out of a desire to perceive the hierarchy of sets as an algebraic structure of a simple kind. We began our work on the hierarchy and on small maps around 1988, simultaneously with related work on open maps (see our (1990), (1994) in the bibliography). The main results were announced in our (1991), and presented in several lectures between 1990 and 1993.

We are most grateful to J. Bénabou, M. Jidbladze and J. van Oosten for their valuable comments on earlier drafts of this book, and to Elise Goeree for the careful typing of the manuscript.

Our work would not have been possible without the support for many mutual visits, by the Canadian National Science and Engineering Research Council and the Netherlands Organization for Scientific Research (NWO).

In this chapter we discuss the solution of problems by a number of processors working in concert. In specifying an algorithm for such a setting, we must specify not only the sequence of actions of individual processors, but also the actions they take in response to the actions of other processors. The organization and use of multiple processors has come to be divided into two categories: parallel processing and distributed processing. In the former, a number of processors are coupled together fairly tightly: they are similar processors running at roughly the same speeds and they frequently exchange information with relatively small delays in the propagation of such information. For such a system, we wish to assert that at the end of a certain time period, all the processors will have terminated and will collectively hold the solution to the problem. In distributed processing, on the other hand, less is assumed about the speeds of the processors or the delays in propagating information between them. Thus, the focus is on establishing that algorithms terminate at all, on guaranteeing the correctness of the results, and on counting the number of messages that are sent between processors in solving a problem. We begin by studying a model for parallel computation. We then describe several parallel algorithms in this model: sorting, finding maximal independent sets in graphs, and finding maximum matchings in graphs. We also describe the randomized solution of two problems in distributed computation: the choice coordination problem and the Byzantine agreement problem.

THE study of random walks on graphs is fascinating in its own right. In addition, it has a number of applications to the design and analysis of randomized algorithms. This chapter will be devoted to studying random walks on graphs, and to some of their algorithmic applications. We start by describing a simple algorithm for the 2-SAT problem, and analyze it by studying the properties of random walks on the line. Following a brief treatment of the basics of Markov chains, we consider random walks on undirected graphs. It is shown that there is a strong connection between random walks and the theory of electric networks. Random walks are then applied to the problem of determining the connectivity of graphs. Next, we turn to the study of random walks on expander graphs. We define a class of expanders and use algebraic graph theory to characterize their properties. Finally, we illustrate the special properties of random walks on expanders via an application to probability amplification.

Let G = (V,E) be a connected, undirected graph with n vertices and m edges. For a vertex v Є V, Γ(v) denotes the set of neighbors of v in G. A random walk on G is the following process, which occurs in a sequence of discrete steps: starting at a vertex v0, we proceed at the first step to a randomly chosen neighbor of V0. This may be thought of as choosing a random edge incident on V0 and walking along it to a vertex v1 Є Γ(v0).

ALL the algorithms we have studied so far receive their entire inputs at one time. We turn our attention to online algorithms, which receive and process the input in partial amounts. In a typical setting, an online algorithm receives a sequence of requests for service. It must service each request before it receives the next one. In servicing each request, the algorithm has a choice of several alternatives, each with an associated cost. The alternative chosen at a step may influence the costs of alternatives on future requests. Examples of such situations arise in data-structuring, resource-allocation in operating systems, finance, and distributed computing.

In an online setting, it is often meaningless to have an absolute performance measure for an algorithm. This is because in most such settings, any algorithm for processing requests can be forced to incur an unbounded cost by appropriately choosing the input sequence (we study examples of this below); thus, it becomes difficult, if not impossible, to perform a comparison of competing strategies. Consequently, we compare the total cost of the online algorithm on a sequence of requests, to the total cost of an offline algorithm that services the same sequence of requests. We refer to such an analysis of an online algorithm as a competitive analysis; we will make these notions formal presently.

IN this chapter we consider several fundamental optimization problems involving graphs: all-pairs shortest paths, minimum cuts, and minimum spanning trees. In each case, deterministic polynomial time algorithms are known, but the use of randomization allows us to obtain significantly faster solutions for these problems. We show that the problem of computing all-pairs shortest paths is reducible, via a randomized reduction, to the problem of multiplying two integer matrices. We present a fast randomized algorithm for the min-cut problem in undirected graphs, thereby providing evidence that this problem may be easier than the max-flow problem. Finally, we present a linear-time randomized algorithm for the problem of finding minimum spanning trees.

Unless stated otherwise, all the graphs we consider are assumed to be undirected and without multiple edges or self-loops. For shortest paths and min-cuts we restrict our attention to unweighted graphs, although in some cases the results generalize to weighted graphs; we give references in the discussion at the end of the chapter.

All-pairs Shortest Paths

Let G(V,E) be an undirected, connected graph with V = ﹛l,…,n﹜ and \E\ = m. The adjacency matrix A is an n x n 0-1 matrix with Aij = Aji = 1 if and only if the edge (i, j) is present in E. Given A, we define the distance matrix D as an n x n matrix with non-negative integer entries such that Dij equals the length of a shortest path from vertex i to vertex j.