A hypercubic sorting network with nearly logarithmic depth

Abstract

A natural class of "hypercubic" sorting networks is defined. The regular structure of these sorting networks allows for elegant and efficient implementations on any of the so-called hypercubic networks (e.g., the hypercube, shuffle-exchange, butterfly, and cube-connected cycles). This class of sorting networks contains Batcher's O(lg2 n)-depth bitonic sort, but not the O(lg n)-depth sorting network of Ajtai, Komlós, and Szemerédi. In fact, no o(lg2 n)-depth compare-interchange sort was previously known for any of the hypercubic networks. In this paper, we prove the existence of a family of 2O(√lg lg n) lg n-depth hypercubic sorting networks. Note that this depth is o(lg1+∈ n) for any constant ∈ > 0.