Parametrized Metrical Task Systems

We consider parametrized versions of metrical task systems and metrical service systems, two fundamental models of online computing, where the constrained parameter is the number of possible distinct requests m. Such parametrization occurs naturally in a wide range of applications. Striking examples are certain power management problems, which are modeled as metrical task systems with m = 2. We characterize the competitive ratio in terms of the parameter m for both deterministic and randomized algorithms on hierarchically separated trees. Our findings uncover a rich and unexpected picture that differs substantially from what is known or conjectured about the unparametrized versions of these problems. For metrical task systems, we show that deterministic algorithms do not exhibit any asymptotic gain beyond one-level trees (namely, uniform metric spaces), whereas randomized algorithms do not exhibit any asymptotic gain even for one-level trees. In contrast, the special case of metrical service systems (subset chasing) behaves very differently. Both deterministic and randomized algorithms exhibit gain, for m sufficiently small compared to n, for any number of levels. Most significantly, they exhibit a large gain for uniform metric spaces and a smaller gain for two-level trees. Moreover, it turns out that in these cases (as well as in the case of metrical task systems for uniform metric spaces with m being an absolute constant), deterministic algorithms are essentially as powerful as randomized algorithms. This is surprising and runs counter to the ubiquitous intuition/conjecture that, for most problems that can be modeled as metrical task systems, the randomized competitive ratio is polylogarithmic in the deterministic competitive ratio

Topology Matters: Smoothed Competitiveness of Metrical Task Systems

We consider online problems that can be modeled as metrical task systems: An online algorithm resides in a graph of n nodes and may move in this graph at a cost equal to the distance. The algorithm has to service a sequence of tasks that arrive over time; each task specifies for each node a request cost that is incurred if the algorithm services the task in this particular node. The objective is to minimize the total request plus travel cost. Borodin, Linial and Saks gave a deterministic work function algorithm (WFA) for metrical task systems having a tight competitive ratio of 2n-1. We present a smoothed competitive analysis of WFA. Given an adversarial task sequence, we add some random noise to the request costs and analyze the competitive ratio of WFA on the perturbed sequence. We prove upper and matching lower bounds. Our analysis reveals that the smoothed competitive ratio of WFA is much better than its (worst case) competitive ratio and that it depends on several topological parameters of the graph underlying the metric, such as maximum degree, diameter, etc. For example, already for moderate perturbations, the smoothed competitive ratio of WFA is O(log(n)) on a clique and O(sqrt{n}) on a line. We also provide the first average case analysis of WFA. For a large class of probability distributions, we prove that WFA has O(log(D)) expected competitive ratio, where D is the maximum degree of the underlying graph

Ramsey-type theorems for metric spaces with applications to online problems

A nearly logarithmic lower bound on the randomized competitive ratio for the metrical task systems problem is presented. This implies a similar lower bound for the extensively studied k-server problem. The proof is based on Ramsey-type theorems for metric spaces, that state that every metric space contains a large subspace which is approximately a hierarchically well-separated tree (and in particular an ultrametric). These Ramsey-type theorems may be of independent interest.Comment: Fix an error in the metadata. 31 pages, 0 figures. Preliminary version in FOCS '01. To be published in J. Comput. System Sc

Randomized online computation with high probability guarantees

We study the relationship between the competitive ratio and the tail distribution of randomized online minimization problems. To this end, we define a broad class of online problems that includes some of the well-studied problems like paging, k-server and metrical task systems on finite metrics, and show that for these problems it is possible to obtain, given an algorithm with constant expected competitive ratio, another algorithm that achieves the same solution quality up to an arbitrarily small constant error a with high probability; the "high probability" statement is in terms of the optimal cost. Furthermore, we show that our assumptions are tight in the sense that removing any of them allows for a counterexample to the theorem. In addition, there are examples of other problems not covered by our definition, where similar high probability results can be obtained.Comment: 20 pages, 2 figure