A Study on Planar 2-Center Problem

Abstract

The Planar-k-Centre problem is an important problem in the class of Optimal Facility Location problems, where given n points in the planar, the objective is finding the smallest k congruent discs such that their union encompasses all points. This dissertation examines a variant of this problem in which the L1 metric is used to determine the distances rather than the standard Euclidean metric, which we call L1P2C and a closely related problem which we call Undirected k Square Coverage where there is no directional constraint for the k squares which contain the given set of points. We have found two deterministic algorithms for the L1P2C problem, one operating in O(n log n) time and the other in O(n) time. Additionally, for k = 1, we have found an O(n log n) time method for the Undirected k Square Coverage problem.