Optimal Schedule for All-to-All Personalized Communication in Multiprocessor Systems

Abstract

In this article, we address the problem of finding an optimal schedule for all-to-all personalized message communication among the processors in a multiprocessor system where every processor has a unique message for every other processor. When there are n processors and ⌊ n /2⌋ parallel databus or channels for message communications, there exist algorithms that require O ( n 2 ) time for assigning the databus/channels to the processor-pairs to obtain a schedule with minimum number of time slots. However, in recent massively parallel processing systems with a huge number of processors, the number k of available databus/channels is usually much smaller than ⌊ n /2⌋. Thus, in each round of communication, only k processor-pairs ( k < ⌊ n /2⌋) can exchange their messages in parallel. We address this general case of all-to-all personalized communication and present a new technique for scheduling the channels among processor-pairs where only k < ⌊ n /2⌋ databus/channels are available. We show that the proposed technique is optimal for all values of n other than 2 k < n < 3 k . For 2 k < n < 3 k , we show that the required number of rounds may be more than the lower bound of ⌈ n /( n −1)2 k ⌉ by at most 11%, and proving the optimality of our technique remains as an open problem in this case.

Realizing Optimal All-to-All Personalized Communication Using Butterfly-Based Networks

“…In [5], the all-to-all personalized communication problem is addressed for parallel systems with a huge number of processors n and a number of data-bus smaller than n/2. A technique for scheduling the channels among processorpairs where only k < ⌊ n 2 ⌋ databus/channels are available is presented.…”

“…All-to-all personalized communication problem has been extensively studied in the past years for many network topologies, such as meshes, tori and hypercubes, and more recently also for hierarchical networks, [4], databus, [5], fat tree [6], [7], and Multistage Interconnection Networks (MINs), in particular, binary MINs. Binary MINs of size N consist of log 2 N stages, each composed of N /2 nodes of size 2 × 2, and present attractive advantages such as efficient routing algorithms, partitionability, small number of very simple switching elements.…”