Delayed path coupling and generating random permutations

We consider the problem of generating permutations almost uniformly at random in distributed and parallel systems. We propose a simple distributed scheme for permuting at random, which we call distributed mixing, and provide its precise stochastic analysis. Our main result is that distributed mixing needs log n simple point-to-point communication rounds to generate a permutation almost uniformly at random. We further apply distributed mixing to design very fast parallel algorithms for OCPC and QRQW parallel computers (with runtimes log log n and √log n respectively). Our analysis of distributed mixing is based on the analysis of the mixing time of the Markov chain governing the process. The main technical tool developed in the paper is a novel method of analyzing convergence of Markov chains. Our method, called delayed path coupling, is a refinement of the classical coupling technique and the path coupling technique of Bubley and Dyer, and its main, novel feature is the use of possible non-Markovian coupling. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 17, 238–259, 2000 Correspondence to: M. Kutyłowski * Part of this work was done while the authors were with the University of Paderborn and the second author was with the University of Wrocław. Research partially supported by KBN grant 8 T11C 032 15, ALCOM EU ESPRIT Long Term Research Project 20244 (ALCOM-IT), DFG-Sonderforschungsbereich 376 “Massive Parallelität”, and DFG Leibniz Grant Me872/6-1. A preliminary report on this research appeared in Delayed path coupling and generating random permutations via distributed stochastic processes, A. Czumaj, P. Kanarek, M. Kutyłowski, and K. Loryś, Proceedings of the 10th ACM-SIAM Symposium on Discrete Algorithms, pp. 271–280, SIAM, 1999. © 2000 John Wiley & Sons, Inc.