Gossiping in vertex-Disjoint paths mode in d-dimensional grids and planar graphs

Abstract

This paper continues with the study of the communication modes introduced by J. Hromkovicč, R. Klasing, E. A. Stö;hr [“Dissemination of Information in Vertex-Disjoint Paths Mode, Part 1: General Bounds and Gossiping in Hypercube-Like Networks,” Technical Report No. tr-ri-93-129, University of Paderborn. (Extended abstract presented at WG’93.)] as a generalization of the standard one-way and two-way modes allowing messages to be sent between processors of interconnection networks via vertex-disjoint paths in one communication step. The complexity of communication algorithms is measured by the number of communication steps (rounds). Here, the complexity of gossiping in grids and in planar graphs is investigated. The main results are the following: 1. Effective one-way and two-way gossip algorithms for d-dimensional grids, d ≥ 2, are designed. 2. The lower bound 2 log2n - log2k - log2 log2n - 4 is established on the number of rounds of every two-way gossip algorithm working on any graph of n nodes and vertex bisection k. This proves that the designed two-way gossip algorithms on d-dimensional grids, d ≥ 3, are almost optimal, and it also shows that the 2-dimensional grid belongs to the best gossip graphs among all planar graphs. 3. Another lower bound proof is developed to get some tight lower bounds on one-way “well-structured” gossip algorithms on planar graphs (to the best of the authors’ knowledge, to date all gossip algorithms designed in vertex-disjoint paths mode have been “well-structured”). © 1995 Academic Press, Inc.