BOUNDED-VELOCITY APPROXIMATION OF MOBILE EUCLIDEAN 2-CENTRES

Abstract

Given a set P of points (clients) in the plane, a Euclidean 2-centre of P is a set of two points (facilities) in the plane such that the maximum distance from any client to its nearest facility is minimized. Geometrically, a Euclidean 2-centre of P corresponds to a cover of P by two discs of minimum radius r (the Euclidean 2-radius). Given a set of mobile clients, where each client follows a continuous trajectory in the plane with bounded velocity, the motion of the corresponding mobile Euclidean 2-centre is not necessarily continuous. Consequently, we consider strategies for defining the trajectories of a pair of mobile facilities that guarantee a fixed-degree approximation of the Euclidean 2-centre while maintaining bounded relative velocity. In an attempt to balance the conflicting goals of closeness of approximation and a low maximum relative velocity, we introduce reflection-based 2-centre functions by reflecting the position of a mobile client across the mobile Steiner centre and the mobile rectilinear 1-centre, respectively.