Cyclic reduction on a binary tree

Abstract

Ensembles of large numbers of processors tightly coupled into networks are of increasing interest. Binary tree interconnect has many favourable characteristics from a construction point of view, though the limited communication bandwidth between arbitrary processors poses a potential bottleneck. In this paper we present an algorithm for odd-even cyclic reduction on a binary tree for which the limited bandwidth does not increase the order of the computational complexity, compared to an ideal parallel machine. The complexity is 2 log 2 N with respect to arithmetic operations, and 3 log 2 N with respect to communication. The communication complexity compares favourably with the best previously published result, O (log 2 2 N). We also show that the benefits of truncated cyclic reduction are much greater for parallel reduction algorithms than for sequential algorithms. A reduction in the computational complexity proportional to the reduction in the number of reduction steps is possible.