Publisher Summary
This chapter discusses parallel algorithms for shared-memory machines. Parallel computation is rapidly becoming a dominant theme in all areas of computer science and its applications. It is estimated that, within a decade, virtually all developments in computer architecture, systems programming, computer applications and the design of algorithms will be taking place within the context of parallel computation. In preparation for this revolution, theoretical computer scientists have begun to develop a body of theory centered on parallel algorithms and parallel architectures. As there is no consensus yet on the appropriate logical organization of a massively parallel computer, and as the speed of parallel algorithms is constrained as much by limits on interprocessor communication as it is by purely computational issues, it is not surprising that a variety of abstract models of parallel computation have been pursued. Closest to the hardware level are the VLSI models, which focus on the technological limits of today's chips, in which gates and wires are packed into a small number of planar layers.

Parallel algorithms for shared-memory machines

The class NC consists of problems solvable very fast (in time polynomial in log n ) in parallel with a feasible (polynomial) number of processors. Many natural problems in NC are known; in this paper an attempt is made to identify important subclasses of NC and give interesting examples in each subclass. The notion of NC 1 -reducibility is introduced and used throughout (problem R is NC 1 -reducible to problem S if R can be solved with uniform log-depth circuits using oracles for S ). Problems complete with respect to this reducibility are given for many of the subclasses of NC . A general technique, the “parallel greedy algorithm,” is identified and used to show that finding a minimum spanning forest of a graph is reducible to the graph accessibility problem and hence is in NC 2 (solvable by uniform Boolean circuits of depth O (log 2 n ) and polynomial size). The class LOGCFL is given a new characterization in terms of circuit families. The class DET of problems reducible to integer determinants is defined and many examples given. A new problem complete for deterministic polynomial time is given, namely, finding the lexicographically first maximal clique in a graph. This paper is a revised version of S. A. Cook, (1983, in “Proceedings 1983 Intl. Found. Comut. Sci. Conf.,” Lecture Notes in Computer Science Vol. 158, pp. 78–93, Springer-Verlag, Berlin/New York).

A taxonomy of problems with fast parallel algorithms

We present a new algorithm for finding a maximum matching in a general graph. The special feature of our algorithm is that its only computationally non-trivial step is the inversion of a single integer matrix. Since this step can be parallelized, we get a simple parallel (RNC 2) algorithm. At the heart of our algorithm lies a probabilistic lemma, the isolating lemma. We show other applications of this lemma to parallel computation and randomized reductions.

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Matching is as easy as matrix inversion

Publisher Summary This chapter discusses the concepts needed for defining the complexity classes. A complexity class is a set of problems of related resource-based complexity. A typical complexity class has a definition of the form—the set of problems that can be solved by an abstract machine M using O(f(n)) of resource R , where n is the size of the input. The simpler complexity classes are defined by various factors. The type of computational problem in which the most commonly used problems are decision problems. However, complexity classes can be defined based on function problems, counting problems, optimization problems, promise problems, etc. The most common model of computation is the deterministic Turing machine, but many complexity classes are based on nondeterministic Turing machines, etc.

A catalog of complexity classes

This volume provides an ideal introduction to key topics in parallel computing. With its cogent overview of the essentials of the subject as well as lists of P -complete- and open problems, extensive remarks corresponding to each problem, a thorough index, and extensive references, the book will prove invaluable to programmers stuck on problems that are particularly difficult to parallelize. In providing an up-to-date survey of parallel computing research from 1994, Topics in Parallel Computing will prove invaluable to researchers and professionals with an interest in the super computers of the future.

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Limits to Parallel Computation: P-Completeness Theory

It is shown that a probabilistic Turing acceptor or transducer running within space bound S can be simulated by a time S2 parallel machine and therefore by a space S2 deterministic machine. (Previous simulations ran in space S6.) In order to achieve these simulations, known algorithms are extended for the computation of determinants in small arithmetic parallel time to computations having small Boolean parallel time, and this development is applied to computing the completion of stochastic matrices. The method introduces a generalization of the ring of integers, called well-endowed rings. Such rings possess a very efficient parallel implementation of the basic (+,−,×) ring operations.

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Parallel computation for well-endowed rings and space-bounded probabilistic machines

It is shown that a probabilistic Turing acceptor or transducer running within space bound S can be simulated by a time S 2 parallel machine and therefore by a space S 2 deterministic machine. (Previous simulations ran in space $6.) In order to achieve these simulations, known algorithms are extended for the computation of determinants in small arithmetic parallel time to computations having small Boolean parallel time, and this development is applied to computing the completion of stochastic matrices. The method introduces a generalization of the ring of integers, called well-endowed rings. Such rings possess a very efficient parallel implementation of the basic (+, -, ×) ring operations.

N. Pippenger

Papers

Parallel computation for well-endowed rings and space-bounded probabilistic machines

Parallel Computation for Well-Endowed Rings