Facility location and the geometric minimum-diameter spanning tree

J. Gudmundsson, H.J. Haverkort, S.M. Park, C.S. Shin, A. Wolff

    Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

    6 Citations (Scopus)

    Abstract

    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 i 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 (n 2 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 /¿ and the O*-notation hides terms of type O (log O(1) E).The other uses the well- separated pair decomposition and takes O (nE 3+En log 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.
    Original languageEnglish
    Title of host publicationApproximation algorithms for combinatorial optimization : proceedings APPROX 2002, Rome Italy, september 17-21, 2002
    EditorsK. Jansen, S. Leonardi, V. Vazirani
    Place of PublicationBerlin
    PublisherSpringer
    Pages146-160
    ISBN (Print)3-540-44186-7
    DOIs
    Publication statusPublished - 2002

    Publication series

    NameLecture Notes in Computer Science
    Volume2462
    ISSN (Print)0302-9743

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