Minimum diameter spanning trees and related problems
SIAM Journal on Computing
Efficiently approximating the minimum-volume bounding box of a point set in three dimensions
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
Approximating the diameter, width, smallest enclosing cylinder, and minimum-width annulus
Proceedings of the sixteenth annual symposium on Computational geometry
Semi-online maintenance of geometric optima and measures
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Farthest-point queries with geometric and combinatorial constraints
Computational Geometry: Theory and Applications
Farthest-point queries with geometric and combinatorial constraints
Computational Geometry: Theory and Applications
Farthest-Point queries with geometric and combinatorial constraints
JCDCG'04 Proceedings of the 2004 Japanese conference on Discrete and Computational Geometry
Minimum-sum dipolar spanning tree in R3
Computational Geometry: Theory and Applications
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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 show that this problem can be solved in O(n2 log n) time and that it yields a factor-4/3 approximation of the MDST.Second, we give two fast approximation schemes for the MDST. One uses a grid and takes O*(E6-1/3 + n) time, where E = 1/驴 and the O*-notation hides terms of type O(logO(1) E). The other uses the well-separated pair decomposition and takes O(nE3+En log n) time. A combination of the two approaches runs in O*(E5 + n) time. Both schemes can also be applied to MSST and 2CP.