Managing Multiple Mobile Resources - Feldkord, Björn; Knollmann, Till; Malatyali, Manuel; Heide, Friedhelm Meyer auf der (Theory of computing systems, 2021)
We extend the Mobile Server problem introduced in Feldkord and Meyer auf der Heide (TOPC 6 (3), 14:1–14:17 2019 ) to a model where k identical mobile resources, here named servers, answer requests appearing at points in the Euclidean space. 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 similarly 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 + δ ) 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 an 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/ δ and the competitive ratio of the simulated k -Page Migration algorithm. We also show how to directly adapt the Double Coverage algorithm (Chrobak et al. SIAM J. Discrete Math. 4 (2), 172–181 11 ) for the k -Server problem to receive an algorithm with improved competitiveness on the line.