Optimal competitiveness for the Rectilinear Steiner Arborescence problem

TL;DR

This paper develops optimal online algorithms for the Rectilinear Steiner Arborescence problem, significantly improving bounds and distinguishing it from related problems, with implications for multimedia content distribution networks.

Contribution

It provides the first tight bounds for the competitive ratio of RSA, separating it from Symmetric-RSA, and extends these results to related multimedia distribution problems.

Findings

Improved bounds on RSA competitive ratio to Θ(log N / log log N)

Separated RSA from Symmetric-RSA in terms of competitive ratios

Established tight bounds for multimedia content distribution problems

Abstract

We present optimal online algorithms for two related known problems involving Steiner Arborescence, improving both the lower and the upper bounds. One of them is the well studied continuous problem of the {\em Rectilinear Steiner Arborescence} ($RSA$). We improve the lower bound and the upper bound on the competitive ratio for $RSA$ from $O(\log N)$ and $\Omega(\sqrt{\log N})$ to $\Theta(\frac{\log N}{\log \log N})$, where $N$ is the number of Steiner points. This separates the competitive ratios of $RSA$ and the Symetric-$RSA$, two problems for which the bounds of Berman and Coulston is STOC 1997 were identical. The second problem is one of the Multimedia Content Distribution problems presented by Papadimitriou et al. in several papers and Charikar et al. SODA 1998. It can be viewed as the discrete counterparts (or a network counterpart) of $RSA$. For this second problem we present tight bounds also in terms of the network size, in addition to presenting tight bounds in terms of the number of Steiner points (the latter are similar to those we derived for $RSA$).