Constant-Time parallel integer sorting

Abstract

We investigate a nonstandard output convention for sorting. Specifically, the input elements are to be returned not in an array sorted in nondecrea.sing order, but in a linked list sorted in nondecreasing order. This problem, which we call chuin-sorting, may be viewed as standard sorting "minus" list ranking, since chain-sorting followed by list ranking of the resulting list is equivalent to standard soI ting. The known lower bounds on parallel integer sorting apply to list ranking, but not to chain-sorting, which is one motivation for considering the latter problem: If chain-sorting constitutes the "essence" of sorting, as indeed we feel, what happens if it is isolated from list ranking? We have the following results for chain-sorting n integers in the range 1. n on a CRC W PRAM: (1) 0(1) time using n2 processors (trivial); (2) O(1) expected time using O(nlog n/log logn) processors (simple); (3) O(log n/log log n) expect ed time using O(n log log n/log n) processors (nont livial); and (4) O(loglognlog n/log loglogn) expected time using O(n log log log n/(log log n log" n)) processors (quite involved). As a by-product of results (3) or (4) above, we are able to improve the best previous results on (standard) randomized sorting of n integers in the range 1. n. The paper introduces several novel techniques in the design of very fast parallel algorithms.