# Facility Location and the Geometric Minimum-Diameter Spanning TreeJ. Gudmundsson, H. Haverkort, S. Park, C. Shin, and A. Wolff. Computational Geometry: Theory and Applications 27 (1): 87--106 (2004)

Let $P$ be a set of $n$ points in the plane. The geometric minimum-diameter spanning tree (MDST) of $P$ is a tree that spans $P$ and minimizes the Euclidian length of the longest path. It is known that there is always a mono- or a dipolar MDST, i.e.\ a MDST with one or two nodes of degree greater 1, respectively. The more difficult dipolar case can so far only be solved in slightly subcubic time. This paper has two aims. First, we present a solution to a new data structure for facility location, the minimum-sum dipolar spanning tree (MSST), that mediates between the minimum-diameter dipolar spanning tree and the discrete two-center problem (2CP) in the following sense: find two centers $p$ and $q$ in $P$ that minimize the sum of their distance plus the distance of any other point (client) to the closer center. This is of interest if the two centers do not only serve their customers (as in the case of the 2CP), but frequently have to exchange goods or personnel between themselves. We give an $O(n^2 łog n)$-time algorithm for this problem. A slight modification of our algorithm yields a factor-4/3 approximation of the MDST. Second, we give two fast approximation schemes for the MDST, i.e.\ factor-($1+\varepsilon$) approximation algorithms. One uses a grid and takes $O^*(E^6-1/3+n)$ time, where $E=1/\varepsilon$ and the $O^*$-notation hides terms of type $O(łog^O(1) E)$. The other uses the well-separated pair decomposition and takes $O(n E^3 + E n n)$ time. A combination of the two approaches runs in $O^*(E^5 + n)$ time. Both schemes can also be applied to MSST and 2CP.
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