Approximate and exact deterministic parallel selection

Abstract

The selection problem of size n is, given a set of n elements drawn from an ordered universe and an integer r with 1 ≤ r ≤ n, to identify the rth smallest dement in the set. We study approximate and exact selection on deterministic concurrent-read concurrent-write parallel RAMs, where approximate selection with relative accuracy λ > 0 asks for any element whose true rank differs from r by at most λn. Our main results are: (1) For all t≥ (log log n)4, approximate selection problems of size n can be solved in O(t) time with optimal speedup with relative accuracy 2-t/(log log n)4 no deterministic PRAM algorithm for approximate selection with a running time below Θ(log n/log log n) was previously known. (2) Exact selection problems of size n can be solved in O(log n/log log n) time with O(n log log n/log n) processors. This running time is the best possible (using only a polynomial number of processors), and the number of processors is optimal for the given running time (optimal speedup); the best previous algorithm achieves optimal speedup with a running time of O(log n log* n/log log n).