Kinetic Facility Location

Abstract

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 optimal cost. 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 requires $\mathcal{O}(n(\log^{d}(n)+\log (nR)))$space in total, where $R:=\frac{\max_{p_{i}\in\mathcal{P}}{f_{i}}\cdot\max_{p_{i}\in\mathcal{P}}{d_{i}}}{\min_{p_{i}\in\mathcal {P}}{f_{i}}\cdot\min_{p_{i}\in\mathcal{P}}{d_{i}}}$, ℘={p 1,p 2,…,p n } is the set of given points, and f i , d i are the maintenance cost and the demand of a point p i , respectively. In case that each trajectory can be described by a bounded degree polynomial, we process $\mathcal{O}(n^{2}\log^{2}(nR))$events, each requiring $\mathcal{O}(\log^{d+1}(n)\cdot\log(nR))$time and $\mathcal {O}(\log(nR))$status changes.