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We show that n integers in the range 1. . n can be stably sorted on an EREW PRAM using O(t) time and o (n(..jlognloglogn+ (logn)2jt)) operations, for arbitrary given t ~ lognloglogn, and on a CREW PRAM using O(t) time and O(n(..jlogn + lognj2 t / 1og ,,)) operations, for arbitrary given t ~ log n. In addition, we are able to sort n arbitrary integers on a randomized CREW PRAM within the same resource bounds with high probability. In each case our algorithm is a factor of almost E>(.y1'Og1i) closer to optimality than an previous algorithms for the stated problem in the stated model, and our third result matches the operation count of the best known sequential algorithm. We also show that n integers in the range 1 .. m can be sorted in O((logn)2) time with O(n) operations on an EREW PRAM using a nonstandard word length of o (log nloglog n log m) bits, thereby greatly improving the upper bound on the word length necessary to sort integers with alinear time-processor product, even sequentially. Our algorithms were inspired by, and in one case directly use, the fusion trees of Ftedman and Willard.

We show that a unit-cost RAM with a word length of w bits can sort n integers in the range 0 } } } 2 w &1 in O(n log log n) time for arbitrary w log n, a significant improvement over the bound of O(n -log n) achieved by the fusion trees of Fredman and Willard. Provided that w (log n) 2+= for some fixed =>0, the sorting can even be accomplished in linear expected time with a randomized algorithm. Both of our algorithms parallelize without loss on a unit-cost PRAM with a word length of w bits. The first one yields an algorithm that uses O(log n) time and O(n log log n) operations on a deterministic CRCW PRAM. The second one yields an algorithm that uses O(log n) expected time and O(n) expected operations on a randomized EREW PRAM, provided that w (log n) 2+= for some fixed =>0. Our deterministic and randomized sequential and parallel algorithms generalize to the lexicographic sorting of multiple-precision integers represented in several words. ]