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Akad~mialKiad5 SpringerVerlag
COMBINATORICA 11 (2) (1991)97122
PARALLEL COMPARISON ALGORITHMS FOR APPROXIMATION
PROBLEMS
N. ALON* and Y. AZAR
ReceivedAugust 22, 1988
Suppose we have n elements from a totally ordered domain, and we are allowed to perform p
parallel comparisons in each time unit ( round). In this paper we determine, up to a constant factor,
the time complexity of several approximation problems in the common parallel comparison tree
model of Valiant, for all admissible values of n, p and e, where e is an accuracy parameter determining
the quality of the required approximation. The problems considered include the approximate
maximum problem, approximate sorting and approximate merging. Our results imply as special
cases, all the known results about the time complexity for parallel sorting, parallel merging and
parallel selection of the maximum (in the comparison model), up to a constant factor. We mention
one very special but representative result concerning the approximate maximum problem; suppose
we wish to find, among the given n elements, one which belongs to the biggest n/2, where in each
round we are allowed to ask n binary comparisons. We show that log* n + O(1) rounds are both
necessary and sufficient in the best algorithm for this problem.
1. Introduction
