Distributed Broadcast Revisited: Towards Universal Optimality

Abstract

This paper revisits the classical problem of multi-message broadcast: given an undirected network G, the objective is to deliver k messages, initially placed arbitrarily in G, to all nodes. Per round, one message can be sent along each edge. The standard textbook result is an $O(D+k)$ round algorithm, where D is the diameter of G. This bound is existentially optimal, which means there exists a graph $G'$ with diameter D over which any algorithm needs $\varOmega (D+k)$ rounds.

In this paper, we seek the stronger notion of optimality—called universal optimality by Garay, Kutten, and Peleg [FOCS’93]—which is with respect to the best possible for graph G itself. We present a distributed construction that produces a k-message broadcast schedule with length roughly within an $\tilde{O}(\log n)$ factor of the best possible for G, after $\tilde{O}(D+k)$ pre-computation rounds.

Our approach is conceptually inspired by that of Censor-Hillel, Ghaffari, and Kuhn [SODA’14, PODC’14] of finding many essentially-disjoint trees and using them to parallelize the flow of information. One key aspect that our result improves is that our trees have sufficiently low diameter to admit a nearly-optimal broadcast schedule, whereas the trees obtained by the algorithms of Censor-Hillel et al. could have arbitrarily large diameter, even up to $\Theta (n)$.