Kinetic Maintenance of Mobile k-Centres on Trees

Given a set P of points (clients) on a weighted tree T, a k-centre of P corresponds to a set of k points (facilities) on T such that the maximum graph distance between any client and its nearest facility is minimized. We consider the mobile k-centre problem on trees. Let C denote a set of n mobile clients, each of which follows a continuous trajectory on a weighted tree T. We establish tight bounds on the maximum relative velocity of the 1-centre and 2-centre of C. When each client in C moves with linear motion along a path on T, the motions of the corresponding 1-centre and 2-centre are piecewise linear; we derive a tight combinatorial bound of Θ(n) on the complexity of the motion of the 1-centre and corresponding bounds of O(n 2 α(n)) and Ω(n²) for a 2-centre, where α(n) denotes the inverse Ackermann function. We describe efficient algorithms for calculating the trajectories of the 1-centre and 2-centre of C: the 1-centre can be found in optimal time O(n log n) and a 2-centre can be found in time O(n² log n). These algorithms lend themselves to implementation within the framework of kinetic data structures. Finally, we examine properties of the mobile 1-centre on graphs and describe an optimal unit-velocit