Approximation algorithm for directed telephone multicast problem

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. The telephone k-multicast problem requires to compute a schedule with minimal number of rounds that delivers a message from a given single processor, that generates the message, to all the processors of a given set T ⊆ V, |T| = k. The processors of V\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(max{log k, log n/log k} · br* + k^1/2) rounds for the directed k-multicast problem, where br* is the value of the optimum solution. This significantly improves the previously best-known approximation ratio of O(k^1/3 · log n · br* + k^2/3) due to [EK03]. We show that our algorithm for the directed multicast problem can be used to derive an algorithm with a similar ratio for the directed minimum poise Steiner arborescence problem, that is, the problem of constructing an arborescence that spans a collection T of terminals, minimizing the sum of height of the arborescence plus maximum out-degree in the arborescence.