A k-Median Based Online Algorithm for the Stochastic k-Server Problem

Abstract

We consider the k-server problem in the random arrival model and present a simple k-median based algorithm for this problem. Let $\sigma =\langle r_1,\ldots , r_n\rangle$ be the sequence of requests. Our algorithm will batch the requests into $\log n$ groups where the $(i+1)^{st}$ group contains requests $\langle r_{2^i +1},\ldots , r_{2^{i+1}}\rangle$. To process the requests of group $i+1$, our algorithm will place the k servers at the k-median centers of the first $2^i$ requests. When a new request of this group arrives, the algorithm will simply assign the server associated with the nearest k-median center to serve it. We show that this simple algorithm, in the random arrival model, has a competitive ratio of at most $O(\alpha )$ and an additive cost of $O(\varDelta k\log n)$, where $\varDelta$ is the diameter of the requests and $\alpha$ is a lower bound on the competitive ratio of any online algorithm in this model.

For our analysis, we use the following fact: In the random arrival model, the expected cost of serving the next request is minimized when servers are located at the k-median of the requests that have not yet arrived (unprocessed requests). But our algorithm instead uses the k-median of the requests seen so far as a proxy. Using existing analysis techniques, we obtain only a large bound on the difference between k-median of the unprocessed requests and that of the processed ones. In particular, in addition to $O(\alpha )$ times the optimal cost, for some $\epsilon > 0$, existing analysis techniques will also give an additive cost of $\epsilon n$ for serving n requests. We present a new analysis to show that when the number of processed and unprocessed requests are of comparable sizes, the cost of serving n requests incurs only an additive cost of $k\varDelta$ (independent of n and significantly better than the previous methods). We then apply this bound for serving each of the $\log n$ groups and obtain an overall bound which is $O(\alpha )$ times the optimal cost with an additive error of $O(k\varDelta \log n)$.

A. D. Friedman and S. Raghvendra were supported under grant NSF-CCF 1464276. A. Adiga was supported by the DTRA CNIMS Contract HDTRA1-11-D-0016-0001.