On the Fermat-Weber Point of a Polygonal Chain and its Generalizations

Article type: Research Article

Authors: Bhattacharya, Bhaswar B.

Affiliations: Indian Statistical Institute, 203 B. T. Road, Kolkata, India. bhaswar.bhattacharya@gmail.com

Note: [] Address for correspondence: Indian Statistical Institute, 203 B. T. Road, Kolkata, India

Abstract: In this paper, we study the properties of the Fermat-Weber point for a set of fixed points, whose arrangement coincides with the vertices of a regular polygonal chain. A k-chain of a regular n-gon is the segment of the boundary of the regular n-gon formed by a set of k (≤ n) consecutive vertices of the regular n-gon. We show that for every odd positive integer k, there exists an integer N(k), such that the Fermat-Weber point of a set of k fixed points lying on the vertices a k-chain of a n-gon coincides with a vertex of the chain whenever n ≥ N(k). We also show that $\lceil$πm(m + 1) - π2/4$\rceil$ ≤ N(k) ≤ $\lfloor$πm(m + 1) + 1$\rfloor$, where k (= 2m + 1) is any odd positive integer. We then extend this result to a more general family of point set, and give an O(hk log k) time algorithm for determining whether a given set of k points, having h points on the convex hull, belongs to such a family.

Keywords: Computational geometry, Facility location, Fermat-Weber Problem, Optimization, Polygons

DOI: 10.3233/FI-2011-406

Journal: Fundamenta Informaticae, vol. 107, no. 4, pp. 331-343, 2011