Sublogarithmic approximation for telephone multicast: Path out of jungle

Consider a network of processors modeled by an n-vertex 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 it 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, whereas the processors of V\T may be left uninformed. The case T = V is called broadcast problem. The telephone multicast and broadcast are basic primitives in distributed computing and computer communication theory. Several approximation algorithms with a polylogarithmic ratio were suggested for these problems, and the upper bound on their approximation threshold stands currently on O(log k) and O(log n), respectively. In this paper we devise an O(log k/log log k)-approximation algorithm for the k-multicast problem, and, consequently, an O(log n/log log n)-approximation algorithm for the broadcast problem. Even stronger than that, whenever an instance of the k-multicast problem admits a schedule of length br*, our algorithm guarantees an approximation ratio of O(log k/log br*). As br* is always at least log k, the ratio of O(log k/log log k) follows. In addition, whenever br* = Ω(k^δ) for some constant δ > 0, we obtain a constant O(1/δ)-approximation ratio for the problem. Regarding the techniques, some previous papers on the subject used the idea of covering the set T of terminals by a forest, and broadcasting the message through this forest. In the current paper we develop a novel technique of covering the set of terminals by a jungle, that is, a collection of trees that are even not necessarily edge-disjoint, which nevertheless possess some other useful properties. The usage of jungles enables to obtain much smaller collections of trees, and this is reflected in the improved approximation ratio. We also derive results regarding the directed and edge-dependent heterogeneous k-multicast problems.