Math. SystemsTheory 24, 101-116 (1991) Mathematical Systems Theory © 1991 Springer-Verlag Now York Inc. A Computer-Assisted Optimal Depth Lower Bound for Nine-Input Sorting Networks* Ian Parberry Department of Computer Science,Penn State University, University Park, PA 16802,USA Abstract. It is demonstrated, using a combination of theoretical and experi- mental computer science, that there is no nine-input sorting network of depth six. If a nine-input sorting network of depth six exists, then there exists one with very special structure. There is an efficient algorithm for constructing and testing comparator networks of this form. This algorithm was implemented and executed on a supercomputer. 1. Introduction Oblivious comparison-based sorting received much attention early in the history of parallel computing, and has continued to be the subject of much research. The central problem, dubbed the Bose-Nelson sorting problem by Floyd and Knuth [6] (after Bose and Nelson [5]), is to devise the most efficient method of sorting n values using a fixed sequence of comparison-swap operations. Many popular sequential sorting algorithms such as mergesort are not oblivious, since the sequence of comparisons performed is not the same for all inputs of any given size. In contrast, bubblesort is oblivious. An oblivious comparison-based algorithm for sorting n values is called an n-input sorting network. One measure of the performance a sorting network is its depth, defined to be the number of parallel steps that the algorithm takes given that in one step any number of disjoint comparison-swap operations can take place simultaneously. Sorting networks * This researchwas supported by NSF Grant CCR-8801659and a ResearchInitiation Grant from the Pennsylvania State University. Author's current address: Department of Computer Sciences, University of North Texas, P.O. Box 13886, Denton, TX 76203-3886. USA. Electronic mail: ian©dept.csci,unt.edu.