The Performance of greedy algorithms for the on-line steiner tree and related problems

We study the on-line Steiner tree problem on a general metric space. We show that the greedy on-line algorithm isO(log((d/z)s))-competitive, wheres is the number of regular nodes,d is the maximum metric distance between any two revealed nodes, andz is the optimal off-line cost. Our results refine the previous known bound 9] and show that AlgorithmSB of Bartalet al. 3] for the on-line file allocation problem isO(log logN)-competitive on anN-node hypercube or butterfly network. A lower bound of (log((d/z)s)) is shown to hold.We further consider the on-line generalized Steiner problem on a general metric space. We show that a class of lazy and greedy deterministic on-line algorithms areO(k· logk)-competitive and no on-line algorithm is better than (logk)-competitive, wherek is the number of distinct nodes that appear in the request sequence.For the on-line Steiner problem on a directed graph, it is shown that no deterministic on-line algorithm is better thans-competitive and the greedy on-line algorithm iss-competitive.A preliminary version of this paper has appeared in theProceedings of the Workshop on Algorithms and Data Structures, 1993, Montréal. The first author's research was partially supported by NSF Grant CCR-9009753, whilst that of the second author was partially supported by NSF Grant DDM-8909660 and a University Fellowship from the Graduate School, Yale University.