On Batcher's merge sorts as parallel sorting algorithms

Abstract

We examine the average running times of Batcher's bitonic merge and Batcher's odd-even merge when they are used as parallel merging algorithms. It has been shown previously that the running time of odd-even merge can be upper bounded by a function of the maximal rank difference for elements in the two input sequences. Here we give an almost matching lower bound for odd-even merge as well as a similar upper bound for (a special version of) bitonic merge. From this follows that the average running time of odd-even merge (bitonic merge) is Θ((n/p)(1+log(1+p ²/n))) (O((n/p)(1+log(1+p ²/n))), resp.) where n is the size of the input and p is the number of processors. Using these results we then show that the average running times of odd-even merge sort and bitonic merge sort are O((n/p) (log n + (log(1 +p²/n))²)), that is, the two algorithms are optimal on the average if $n \geqslant p^2 /2^{\sqrt {\log p} }$.