Kinetic Maintenance of Mobile k-Centres in Trees

Given a set P of points (clients) on a weighted tree T , the 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 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(nα(n)) and Ω(n) for the 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) when the distance function between mobile clients is known or O(n) when the function must be calculated, and a 2-centre can be found in time O(n log n). These algorithms lend themselves to implementation within the framework of kinetic data structures, resulting in structures that are compact, efficient, responsive, and local.