Mathematics – Metric Geometry
Scientific paper
2010-04-17
Revised and expanded version of the paper is in Fundamenta Informaticae, Volume 107 (4), pp. 331-343, 2011
Mathematics
Metric Geometry
10 pages, 4 figures.
Scientific paper
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 $k ~(\leq n)$ 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 $n\geq N(k)$. We also show that $N(k)=O(k^2)$ and give an $O(\log k)$ time algorithm for computing $N(k)$. 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.
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