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Abstract

We consider online problems that can be modeled as \emph{metrical task systems}: An online algorithm resides in a graph $G$ of $n$ nodes and may move in this graph at a cost equal to the distance. The algorithm has to service a sequence of \emph{tasks} that arrive online; each task specifies for each node a \emph{request cost} that is incurred if the algorithm services the task in this particular node. The objective is to minimize the total request cost plus the total travel cost. Several important online problems can be modeled as metrical task systems. Borodin, Linial and Saks \cite{BLS92} presented a deterministic \emph{work function algorithm} (WFA) for metrical task systems having a tight competitive ratio of $2n-1$. However, the competitive ratio often is an over-pessimistic estimation of the true performance of an online algorithm. In this paper, we present a \emph{smoothed competitive analysis} of WFA. Given an adversarial task sequence, we smoothen the request costs by means of a symmetric additive smoothing model and analyze the competitive ratio of WFA on the smoothed task sequence. Our analysis reveals that the smoothed competitive ratio of WFA is much better than $O(n)$ and that it depends on several topological parameters of the underlying graph $G$, such as the minimum edge length $U_{\min}$, the maximum degree $D$, and the edge diameter $diam$. Assuming that the ratio between the maximum and the minimum edge length of $G$ is bounded by a constant, the smoothed competitive ratio of WFA becomes $O(diam (U_{\min}/\sigma + \log(D)))$ and $O(\sqrt{n \cdot (U_{\min}/\sigma + \log(D))})$, where $\sigma$ denotes the standard deviation of the smoothing distribution. For example, already for perturbations with $\sigma = \Theta(U_{\min})$ the competitive ratio reduces to $O(\log n)$ on a clique and to $O(\sqrt{n})$ on a line. We also prove that for a large class of graphs these bounds are asymptotically tight. Furthermore, we provide two lower bounds for any arbitrary graph. We obtain a better bound of $O(\beta \cdot (U_{\min}/\psigma + \log(D)))$ on the smoothed competitive ratio of WFA if each adversarial task contains at most $\beta$ non-zero entries. Our analysis holds for various probability distributions, including the uniform and the normal distribution. We also provide the first average case analysis of WFA. We prove that WFA has $O(\log(D))$ expected competitive ratio if the request costs are chosen randomly from an arbitrary non-increasing distribution with standard deviation.