Sorting in linear time?

We show that a unit-cost RAM with a word length of bits can sort integers in the range in time, for arbitrary ! , a significant improvement over the bound of " # $ achieved by the fusion trees of Fredman and Willard. Provided that % & ' ( *),+., for some fixed /102 , the sorting can even be accomplished in linear expected time with a randomized algorithm. Both of our algorithms parallelize without loss on a unitcost PRAM with a word length of bits. The first one yields an algorithm that uses 3 4 5 $ time and 6 ( operations on a deterministic CRCW PRAM. The second one yields an algorithm that uses ' 5 7 expected time and " expected operations on a randomized EREW PRAM, provided that 8 ' 5 7 *),+.for some fixed /90: . Our deterministic and randomized sequential and parallel algorithms generalize to the lexicographic sorting problem of sorting multiple-precision integers represented in several words.

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