Sorting in Linear Time?

We show that a unit-cost RAM with a word length ofwbits can sortnintegers in the range 0?2w?1 inO(nloglogn) time for arbitraryw?logn, a significant improvement over the bound ofO(nlogn) achieved by the fusion trees of Fredman and Willard. Provided thatw?(logn)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 ofwbits. The first one yields an algorithm that usesO(logn) time andO(nloglogn) operations on a deterministic CRCW PRAM. The second one yields an algorithm that usesO(logn) expected time andO(n) expected operations on a randomized EREW PRAM, provided thatw?(logn)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.

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