The Kinetic Facility Location Problem

We present a deterministic kinetic data structure for the facility location problem that maintains a subset of the moving points as facilities such that, at any point of time, the accumulated cost for the whole point set is at most a constant factor larger than the current optimal cost. At any point of time, the cost that arise for a point depends on the status and the position of the point. In the case that a point is currently a facility it causes maintenance cost, otherwise it causes connection cost depending on its demand and its distance to the nearest facility. In our scenario, each point can change its status between client and facility and moves continuously along a known trajectory in a d-dimensional Euclidean space, where d is a constant. Our kinetic data structure needsO(n(log(n)+log(nR))) space, where R := maxpi∈P fi ·maxpi∈P di minpi∈P fi ·minpi∈P di , P = {p1, p2, . . . , pn} is the set of given points, and fi, di are the maintenance cost and the demand of a point pi, respectively. In the case that each trajectory can be described by a bounded degree polynomial, the data structure processes O(n2 log(nR)) events, each requiring O(log(nR)) status changes and O(log(n) · log(nR)) time. This results in a total processing time of O(n2 log(n) · log(nR)). To the best of our knowledge, this is the rst kinetic data structure for the facility location problem.