We extend the Mobile Server Problem, introduced in SPAA'17, to a model where k identical mobile resources, here named servers, answer requests appearing at points in the Euclidean space. In order to reduce communication costs, the positions of the servers can be adapted by a limited distance m_s per round for each server. The costs are measured similar to the classical Page Migration Problem, i.e., answering a request induces costs proportional to the distance to the nearest server, and moving a server induces costs proportional to the distance multiplied with a weight D. We show that, in our model, no online algorithm can have a constant competitive ratio, i.e., one which is independent of the input length n, even if an augmented moving distance of (1+ )m_s is allowed for the online algorithm. Therefore we investigate a restriction of the power of the adversary dictating the sequence of requests: We demand locality of requests, i.e., that consecutive requests come from points in the Euclidean space with distance bounded by some constant m_c. We show constant lower bounds on the competitiveness in this setting (independent of n, but dependent on k, m_s and m_c). On the positive side, we present a deterministic online algorithm with bounded competitiveness when augmented moving distance and locality of requests is assumed. Our algorithm simulates any given algorithm for the classical k-Page Migration problem as guidance for its servers and extends it by a greedy move of one server in every round. The resulting competitive ratio is polynomial in the number of servers k, the ratio between m_c and m_s, the inverse of the augmentation factor 1 and the competitive ratio of the simulated k-Page Migration algorithm.
Abstract The k-server problem is that of planning the motion of k mobile servers on the vertices of a graph under a sequence of requests for service. Each request consists of the name of a vertex, and is satisfied by placing a server at the requested vertex. The requests must be satisfied in their order of occurrence. The cost of satisfying a sequence of requests is the distance moved by the servers. In this paper we study on-line algorithms for this problem from the competitive point of view. That is, we seek to develop on-line algorithms whose performance on any sequence of requests is as close as possible to the performance of the optimum off-line algorithm. We obtain optimally competitive algorithms for several important cases. Because of the flexibility in choosing the distances in the graph and the number of servers, the k -server problem can be used to model a number of important paging and caching problems. It can also be used as a building block for solving more general problems. We show how server algorithms can be used to solve a seemingly more general class of problems known as task systems .