Sublogarithmic approximation for telephone multicast

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 {set minus} T may be left uninformed. The case T = V is called broadcast problem. 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 ( frac(log k, log log k) )-approximation algorithm for the k-multicast problem, and, consequently, an O ( frac(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 ( frac(log k, log br^*) ). As br^* is always at least log k, the ratio of O ( frac(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. Our results have implications for network design. The poise of a spanning tree is the sum of its depth and maximum degree [R. Ravi, Rapid rumor ramification: Approximating the minimum broadcast time, in: Proceedings of the IEEE Symposium on Foundations of Computer Science, FOCS '94, 1994, pp. 202-213]. We improve the O ( log k ) approximation algorithm [A. Bar-Noy, S. Guha, J. Naor, B. Schieber, Multicasting in heterogeneous networks, SIAM J. Comput. 30 (2) (2000) 347-358] for the poise problem to O ( log k / log k log k ), and obtain an improved ( O ( log k / log log k ), O ( log k / log k log k ) ) bicriteria approximation for the depth-degree problem. We also derive results concerning the edge-dependent heterogeneous k-multicast problem.