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.