The relationship between the gossip complexity in vertex-disioint Daths mode and the vertex J bisekion width*

The one-way and two-way communication modes used for sending messages to processors of interconnection networks via vertex-disjoint paths in one communication step are investigated. The complexity of communication algorithms is measured by the number of communication steps (rounds). This paper reveals a direct relationship between the gossip complexity and the vertex bisection width. More precisely, the main results are the following: We prove that for any 2-way gossip algorithm running on a graph Gn,k of n nodes and vertex bisection width k, the lower bound on the number of rounds is 2 log, n-log, k-log, log, k-6. We prove that there is a graph G,,k of n nodes and vertex bisection width k such that there exists a 2-way gossip algorithm for it running in 2 log, n log, k log, log, k + 7 rounds. We prove that for any l-way “well-structured” gossip algorithm running on a graph G,,,k of n nodes and vertex bisection width k, the lower bound on the number of rounds is 2 log, n 0.56. . (log, k + log, log, k) 0( 1) (to the best of the authors’ knowledge, to date all gossip algorithms designed in vertex-disjoint paths mode have been “well-structured”). We prove that there is a graph Gn,k of n nodes and vertex bisection width k such that there exists a l-way gossip algorithm for it running in 2 log, rz-0.56.. :(log, k+log, log, k)+O( I ) rounds. These results improve the previous results and prove their optimality with respect to the class of graphs G,,k of n nodes and vertex bisection width k. Tight lower bounds on one-way “well-structured” gossip algorithms in d-dimensional grids, d>3, several results for gossiping in graphs of constant degree and planar graphs, as well as similar results for the related edge-disjoint paths mode follow.