On Probabilistic Networks for Selection, Merging, and Sorting

Abstract

We study comparator networks for selection, merging, and sorting that output the correct result with high probability, given a random input permutation. We prove tight bounds, up to constant factors, on the size and depth of probabilistic (n,k)-selection networks. In the case of (n, n/2)-selection, our result gives a somewhat surprising bound of $\Theta(n \log \log n)$ on the size of networks of success probability in $[\delta, 1-1/\mbox{poly}(n)]$ , where δ is an arbitrarily small positive constant, thus comparing favorably with the best previously known solutions, which have size $\Theta(n\log n)$ . We also prove tight bounds, up to lower-order terms, on the size and depth of probabilistic merging networks of success probability in $[\delta, 1-1/\mbox{poly}(n)]$ , where δ is an arbitrarily small positive constant. Finally, we describe two fairly simple probabilistic sorting networks of success probability at least $1-1/\mbox{poly}(n)$ and nearly logarithmic depth.