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Probability and Algorithms
shared-memory multiprocessors and fixed interconnection networks. For a survey of PRAM algorithms, both deterministic and randomized, see Karp and Ramachandran (1990).
The PRAM consists of a collection of independent sequential machines, each of which is called a processor, that communicate with one another through a global memory. This is a synchronous model, and a step of a PRAM consists of a read cycle in which each processor can read a global memory cell, a computing cycle in which each processor can perform a unit-time sequential computation, and a write cycle in which each processor can write into a global memory cell. There are many variants of this model, differing in whether a read conflict or a write conflict is allowed and, in the latter case, differing by the method used to resolve a write conflict. Because efficient simulations of the various models are known, we shall not elaborate on these variants.
Parallel algorithms have been designed on the PRAM for a large number of important problems. This has been a rich and exciting area of research in recent years, and many new techniques and paradigms have been developed. However, in spite of impressive gains, some problems have proved to be resistant to attempts to design highly parallel algorithms for their solution. For some other problems, parallel algorithms with good performance have come at the expense of extremely complex and intricate algorithms. In this context, several researchers have turned to randomization in an attempt to obtain better algorithms for these problems.
A randomized parallel algorithm is an algorithm in which each processor has access to a random number generator. The goal is to use this capacity to generate random numbers to come up with an algorithm that solves a problem quickly and with high probability on every input. The requirement that the algorithm achieves its performance guarantee on every input is much stronger than that of demanding good average-case performance and is a highly desirable property in an algorithm that uses randomization. Correspondingly, such algorithms are more difficult to design. Fortunately, there have been a number of successes in the design of such algorithms for parallel machines.
10.2Quality Measures for Randomized Parallel Algorithms
For the purpose of this exposition, it is convenient to consider a problem as a binary relation s(x, y), where x is the input to the problem and y is a possible output for x. The value of s(x, y) will be true if and only if y is a