Modelling gateway placement in wireless networks: Geometric k-centres of unit disc graphs

Abstract

Motivated by the gateway placement problem in wireless networks, we consider the geometric k-centre problem on unit disc graphs: given a set of points P in the plane, find a set F of k points in the plane that minimizes the maximum graph distance from any vertex in P to the nearest vertex in F in the unit disc graph induced by P ∪ F . We show that the vertex 1-centre provides a 7-approximation of the geometric 1-centre and that a vertex k-centre provides a 13-approximation of the geometric k-centre, resulting in an O ( k n ) -time 26-approximation algorithm. We describe O ( n 2 m ) -time and O ( n 3 ) -time algorithms, respectively, for finding exact and approximate geometric 1-centres, and an O ( m n 2 k ) -time algorithm for finding a geometric k-centre for any fixed k. We show that the problem is NP-hard when k is an arbitrary input parameter. Finally, we describe an O ( n ) -time algorithm for finding a geometric k-centre in one dimension.