Kinetic maintenance of mobile image-centres on trees Original Research Article

Given a set image of points (clients) on a weighted tree image, a image-centre of image corresponds to a set of image points (facilities) on image such that the maximum graph distance between any client and its nearest facility is minimised. We consider the mobile image-centre problem on trees. Let image denote a set of image mobile clients, each of which follows a continuous trajectory on a weighted tree image. We establish tight bounds on the maximum relative velocity of the 1-centre and 2-centre of image. When each client in image moves with linear motion along a path on image, the motions of the corresponding 1-centre and 2-centre are piecewise linear; we derive a tight combinatorial bound of image on the complexity of the motion of the 1-centre and corresponding bounds of image and image for a 2-centre, where image denotes the inverse Ackermann function. We describe efficient algorithms for calculating the trajectories of the 1-centre and 2-centre of image: the 1-centre can be found in optimal time image and a 2-centre can be found in time image. 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-velocity 2-approximation.