On the Size of Permutation Networks and Consequences for Efficient Simulation of Hypercube Algorithms on Bounded-Degree Networks

• Published in: SIAM journal on discrete mathematics Vol. 23; no. 3; pp. 1612 - 1645
• Main Authors: Hromkovič, Juraj, Kanarek, PrzemysŁawa, Klasing, Ralf, Loryś, Krzysztof, Unger, Walter, Wagener, Hubert
• Format: Journal Article
• Language: English
• Published: Philadelphia Society for Industrial and Applied Mathematics 01.01.2009
• Subjects: Algorithms; Asymptotic properties; Communication; Communications networks; Computer Science; Computer simulation; Data Structures and Algorithms; Discrete Mathematics; Distributed, Parallel, and Cluster Computing; Gossip; Hypercubes; Mathematical analysis; Networks; Optimization; Permutations; Simulation; 94C15; Parallel algorithms AMS subject classifications. 68M07; 90B18; 68M10; Network design and communication; Communication networks; Permutation networks; Switching networks
• ISSN: 0895-4801; 1095-7146
• DOI: 10.1137/060669164

The sizes of permutation networks and planar permutation networks for special sets of permutations are investigated. Several asymptotically optimal estimations for distinct subsets of the set of all permutations are established here. The two main results are as follows: A consequence of our results is the construction of a 4-degree network which can simulate each communication step of any hypercube algorithm using edges from at most a constant number of different dimensions in one communication step in $O(\log\log N)$ communication steps. An essential improvement of gossiping in vertex-disjoint path mode in bounded-degree networks follows.