Periodic constant depth sorting networks

Abstract

Comparator networks of constant depth can be used for sorting in the following way. The computation consists of a number of iterations, say t, each iteration being a single run through the comparator network. The output of iteration j (j < t) is used as the input for iteration j+1. The output of the iteration t is the output of the computation. In such a way, it is possible to apply a network with a small number of comparators for sorting long input sequences. However, it is not clear how to make such a computation fast.

Odd-Even Transposition Sort gives a periodic sorting network of depth 2, that sorts n numbers in n/2 iterations. The network of depth 8 proposed by Schwiegelshohn [8] sorts n numbers in O(√nlog n) iterations. Krammer

For a fixed but arbitrary k ∃ ℕ, we present a periodic sorting network of depth O(k) that sorts n input numbers in O(k² · n^1/k) steps.

supported by KBN grant 2 1197 91 01 and Volkswagen Foundation, Project “Paralleles Rechnen: Theoretische und experimentelle Untersuchungen zu parallelen Rechnenmodellen und systemnahen Algorithmen”, partially this work was done while the first and the second author visited Heinz-Nixdorf-Institut, Universität Paderborn