Optimal broadcasting and gossiping in one-port meshes of trees with distance-insensitive routing

Abstract

Meshes of trees are an ingenious hybrid of meshes and trees. They have very interesting properties, namely small degree and diameter and large bisection width. Also they are area universal, i.e., they can simulate any network with the same VLSI wire area with only polylogarithmic factor of slowdown. They are known to outperform meshes in execution of algorithms with local communication patterns, for which distance-sensitive switching, such as store-and-forward, suffices. Nowadays, parallel machines use distance-insensitive switching, such as wormhole. The performance of a machine with such a fast switching depends on the existence of efficient algorithms for commonly used collective communication operations. Little is known about collective communication properties of meshes of trees with distance-insensitive switching. We show that 2-D meshes of trees with distance-insensitive switching can execute optimally or asymptotically optimally two basic global communication operations, one-to-all broadcast and all-to-all broadcast, if routers with one-port and packet combining capabilities are used. Our algorithms work optimally for both square and rectangular meshes of trees.

Introduction

The meshes of trees [15] have been proposed as an ingenious hybrid of trees and meshes. They have both small diameter and large bisection width. Many problems can be solved on them in polylogarithmic parallel time by algorithms that use mostly local communication patterns. For these algorithms, meshes of trees clearly outperform meshes of the same size. To implement such algorithms, a distance-sensitive switching hardware, such as store-and-forward, is sufficient. Current parallel computers use distance-insensitive switching, such as wormhole [16], which can, of course, mimic local communication patterns. How meshes of trees compare to meshes if algorithms with global communication patterns are considered, has not been studied yet. In this paper, we show that meshes of trees with distance-insensitive switching and with routers with one-port capability and packet combining can execute optimally or asymptotically optimally two basic collective communication operations [5], [6], [16], one-to-all broadcast (OAB) and all-to-all broadcast (AAB). We consider both node-disjoint-path and link-disjoint-path models [3], [7]. Our algorithms work optimally for both square and rectangular meshes of trees.

This result indicates that meshes of trees have outstanding properties even if distance-insensitive switching hardware is used. Hence, it indicates that as general purpose topology, meshes of trees outperform meshes, since they execute faster algorithms with local communication patterns and do not lag behind meshes in execution of algorithms with global communication patterns.