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

Authors:
S. Durocher;K. R. Jampani;A. Lubiw;L. Narayanan
Affiliations:
Department of Computer Science, University of Manitoba, Winnipeg, Canada;Cheriton School of Computer Science, University of Waterloo, Waterloo, Canada;Cheriton School of Computer Science, University of Waterloo, Waterloo, Canada;Department of Computer Science and Software Engineering, Concordia University, Montréal, Canada
Venue:
Computational Geometry: Theory and Applications
Year:
2011

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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(kn)-time 26-approximation algorithm. We describe O(n^2m)-time and O(n^3)-time algorithms, respectively, for finding exact and approximate geometric 1-centres, and an O(mn^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.