Optimal Prefix Circuits with Fan-out 2

Abstract

Given n values v1,v2,…,vn and an associative binary operation, denoted by o , the prefix problem is to compute the n prefixed v1 o v2o … o vi, 1≦ i ≦ n. Because of the importance of prefix computation, many combinational circuits for solving the prefix problem, called prefix circuits, have been designed. It has been proved that the size s(n) and the depth d(n) of an n-input prefix circuit G(n) satisfy the inequality d(n)+s(n)≧2n-2；thus, a prefix circuit is optimal if d(n)+s(n)=2n-2. For the first time in this paper, we present a systematic method to construct optimal parallel prefix circuits with fan-out 2. We also solve an open problem by building a class of optimal prefix circuits with fan-out 2 whose depth can be any integer between n-1 and 2 ﹝lg n﹞-1,or 2﹝lg n﹞ depending on the value of n .The optimal prefix circuits have corresponding optimal prefix algorithms running on a fully connected message-passing multicomputer.