MPI-I-95-1-024

We show that a unit-cost RAM with a word length of $w$ bits can sort $n$ integers in the range $0\Ttwodots 2^w-1$ in $O(n\log\log n)$ time, for arbitrary $w\ge\log n$, a significant improvement over the bound of $O(n\sqrt{\log n})$ achieved by the fusion trees of Fredman and Willard. Provided that $w\ge(\log n)^{2+\epsilon}$ for some fixed $\epsilon>0$, 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\break $O(n\log\log n)$ 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 $w\ge(\log n)^{2+\epsilon}$ for some fixed $\epsilon>0$. Our deterministic and randomized sequential and parallel algorithms generalize to the lexicographic sorting problem of sorting multiple-precision integers represented in several words.