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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 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 $\Theta((n/p)(1+\log(1+p^2/n)))$ ($O((n/p)(1+\log(1+p^2/n)))$, 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 $O((n/p)(\log n + (\log(1+p^2/n))^2))$,
that is, the two algorithms are optimal on the average if
$n\geq p^2/2^{\sqrt{\log p}}$.
The derived bounds do not allow to compare the two sorting algorithms
program, for various sizes of input and numbers of processors.
