Online Priority Steiner Tree Problems

Abstract

A central issue in the design of modern communication networks is the provision of Quality-of-Service (QoS) guarantees at the presence of heterogeneous users. For instance, in QoS multicasting, a source needs to efficiently transmit a message to a set of receivers, each requiring support at a different QoS level (e.g., bandwidth). This can be formulated as the Priority Steiner tree problem: Here, each link of the underlying network is associated with a priority value (namely the QoS level it can support) as well as a cost value. The objective is to find a tree of minimum cost that spans all receivers and the source, such that the path from the source to any given receiver can support the QoS level requested by the said receiver. The problem has been studied from the point of view of approximation algorithms.

In this paper we introduce and address the on-line variant of the problem, which models the situation in which receivers join the multicast group dynamically. Our main technical result is a tight bound on the competitive ratio of $\Theta \left (\min \left \{b \log \frac{k}{b},k \right \} \right)$ (when k > b), and Θ(k) (when k ≤ b), where b is the total number of different priority values and k is the total number of receivers. The bound holds for undirected graphs, and for both deterministic and randomized algorithms. For the latter class, the techniques of Alon et al. [Trans. on Algorithms 2005] yield a O(logk logm)-competitive randomized algorithm, where m is the number of edges in the graph. Last, we study the competitiveness of online algorithms assuming directed graphs; in particular, we consider directed graphs of bounded edge-cost asymmetry.