Kinetic maintenance of mobile [formula omitted]-centres on trees

Abstract

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 minimised. 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 ( n 2 ) 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 ( n 2 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-velocity 2-approximation.