On Batcher's Merge Sorts as Parallel Sorting Algorithms

In this paper 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 diierence 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 ((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 used. Using these results we then show that the average running times of odd-even merge sort and bitonic merge sort are O((n=p)(logn + (log(1 + p 2 =n)) 2)), that is, the two algorithms are optimal on the average if n p 2 =2 p log p. The derived bounds do not allow to compare the two sorting algorithms directly, thus we also present experimental results, obtained by a simulation program, for various sizes of input and numbers of processors.