Optimal Competitiveness for Symmetric Rectilinear Steiner Arborescence and Related Problems

We present optimal competitive algorithms for two interrelated known problems involving Steiner Arborescence. One is the continuous problem of the Symmetric Rectilinear Steiner Arborescence (SRSA), studied by Berman and Coulston as a symmetric version of the known Rectilinear Steiner Arborescence (RSA) problem. A very related, but discrete problem (studied separately in the past) is the online Multimedia Content Delivery (MCD) problem on line networks, presented originally by Papadimitriu, Ramanathan, and Rangan. An efficient content delivery was modeled as a low cost Steiner arborescence in a grid of network×time they defined. We study here the version studied by Charikar, Halperin, and Motwani (who used the same problem definitions, but removed some constraints on the inputs). The bounds on the competitive ratios introduced separately in the above papers are similar for the two problems: O(logN) for the continuous problem and O(log n) for the network problem, where N was the number of terminals to serve, and n was the size of the network. The lower bounds were Ω( √ logN) and Ω( √ logn) correspondingly. Berman and Coulston conjectured that both the upper bound and the lower bound could be improved. We disprove this conjecture and close these quadratic gaps for both problems. We first present an O( √ log n) deterministic competitive algorithm for MCD on the line, matching the lower bound. We then translate this algorithm to become a competitive optimal algorithm O( √ logN) for SRSA. Finally, we translate the latter back to solve MCD problem, this time competitive optimally even in the case that the number of requests is small (that is, O(min{√logn,√logN})). We also present a Ω( 3 √ logn) lower bound on the competitiveness of any randomized algorithm. Some of the techniques may be useful in other contexts. (For example, rather than comparing to the unknown optimum, we compared the costs of the online algorithm to the costs of an approximation offline algorithm).