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**An approximation algorithm for the directed telephone multicast problem.**
*(English)*
Zbl 1099.68011

Summary: Consider a network of processors modeled by an \(n\)-vertex directed graph \(G = (V,E)\). Assume that the communication in the network is synchronous, i.e., occurs in discrete “rounds”, and in every round every processor is allowed to pick one of its neighbors, and to send him a message. A set of terminals \(\mathcal T \subseteq V\) of size \(|\mathcal T| = k\) is given. The telephone \(k\)-multicast problem requires computing a schedule with a minimal number of rounds that delivers a message from a given single processor, that generates the message, to all the processors of \(\mathcal T\). The processors of \(V\backslash \mathcal T\) may be left uninformed. The telephone multicast is a basic primitive in distributed computing and computer communication theory. In this paper we devise an algorithm that constructs a schedule with \(O(\log k\cdot b^{*}+ k^{1/2})\) rounds for the directed \(k\)-multicast problem, where \(b^{*}\) is the value of the optimum solution. This is the first algorithm with a non-trivial approximation guarantee for this problem. We show that our algorithm for the directed multicast problem can be used to derive an algorithm with a similar ratio for the directed Steiner poise problem, that is, the problem of constructing an arborescence that spans a collection \(\mathcal T\) of terminals and has the minimum poise.