Distributed algorithms in synchronous broadcasting networks

In this paper we consider a synchronous broadcasting network, a distributed computation model which represents communication networks that are used extensively in practice. We consider a basic problem of information sharing: the computation of the multiple identification function. That is, given a network of p processors, each of which contains an n-bit string of information, how can every processor compute efficiently the subset of processors which have the same information as itself? The problem was suggested by Yao as a generalization of the two-processor case studied in his classic paper on distributed computing (Yao, 1979). The naive way to solve this problem takes O(np) communication time, where a time unit is the time to transfer one bit. We present an algorithm which takes advantage of properties of strings and is O(n log² p + p) time. A simulation of sorting networks by the distributed model yields an O(n log p + p) (impractical) algorithm. By applying Yao's probabilistic implementation of the two-processor case to both algorithm we get probabilistic versions (with small error) where n is replaced by log n in the complexity expressions. We also present lower bounds for the problem: an Ω(n) and an Ω(p) bound are shown.