Improved bounds for the online steiner tree problem in graphs of bounded edge-asymmetry

In this paper we consider the Online Steiner Tree problem in weighted directed graphs of bounded edge-asymmetry α. The edge-asymmetry of a directed graph is defined as the maximum ratio of the cost (weight) of antiparallel edges in the graph. The problem has applications in multicast routing over a network with non-symmetric links. We improve the previously known upper and lower bounds on the competitive ratio of any deterministic algorithm due to Faloutsos <i>et al.</i> [11]. In particular, we show that a better analysis of a simple greedy algorithm yields a competitive ratio of <i>O</i> (min {<i>k</i>, α log <i>k</i>/log log α}), where <i>k</i> denotes the number of terminals requested. On the negative side, we show a lower bound of Ω(min{<i>k</i>^1-ε, α log <i>k</i>/log log <i>k</i>}) on the competitive ratio of every deterministic algorithm for the problem, for any arbitrarily small constant ε.