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DOI: 10.4230/LIPIcs.SoCG.2017.4

Minimum Perimeter-Sum Partitions in the Plane

Authors: Mikkel Abrahamsen, Mark de Berg, Kevin Buchin, Mehran Mehr, and Ali D. Mehrabi

Published in: LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

Abstract

Let P be a set of n points in the plane. We consider the problem of partitioning P into two subsets P_1 and P_2 such that the sum of the perimeters of CH(P_1) and CH(P_2) is minimized, where CH(P_i) denotes the convex hull of P_i. The problem was first studied by Mitchell and Wynters in 1991 who gave an O(n^2) time algorithm. Despite considerable progress on related problems, no subquadratic time algorithm for this problem was found so far. We present an exact algorithm solving the problem in O(n log^4 n) time and a (1+e)-approximation algorithm running in O(n + 1/e^2 log^4(1/e)) time.

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Mikkel Abrahamsen, Mark de Berg, Kevin Buchin, Mehran Mehr, and Ali D. Mehrabi. Minimum Perimeter-Sum Partitions in the Plane. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 4:1-4:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{abrahamsen_et_al:LIPIcs.SoCG.2017.4,
  author =	{Abrahamsen, Mikkel and de Berg, Mark and Buchin, Kevin and Mehr, Mehran and Mehrabi, Ali D.},
  title =	{{Minimum Perimeter-Sum Partitions in the Plane}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{4:1--4:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-038-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{77},
  editor =	{Aronov, Boris and Katz, Matthew J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.4},
  URN =		{urn:nbn:de:0030-drops-72048},
  doi =		{10.4230/LIPIcs.SoCG.2017.4},
  annote =	{Keywords: Computational geometry, clustering, minimum-perimeter partition, convex hull}
}

Document

DOI: 10.4230/LIPIcs.SoCG.2017.5

Range-Clustering Queries

Authors: Mikkel Abrahamsen, Mark de Berg, Kevin Buchin, Mehran Mehr, and Ali D. Mehrabi

Published in: LIPIcs, Volume 77, 33rd International Symposium on Computational Geometry (SoCG 2017)

Abstract

In a geometric k-clustering problem the goal is to partition a set of points in R^d into k subsets such that a certain cost function of the clustering is minimized. We present data structures for orthogonal range-clustering queries on a point set S: given a query box Q and an integer k > 2, compute an optimal k-clustering for the subset of S inside Q. We obtain the following results. * We present a general method to compute a (1+epsilon)-approximation to a range-clustering query, where epsilon>0 is a parameter that can be specified as part of the query. Our method applies to a large class of clustering problems, including k-center clustering in any Lp-metric and a variant of k-center clustering where the goal is to minimize the sum (instead of maximum) of the cluster sizes. * We extend our method to deal with capacitated k-clustering problems, where each of the clusters should not contain more than a given number of points. * For the special cases of rectilinear k-center clustering in R^1, and in R^2 for k = 2 or 3, we present data structures that answer range-clustering queries exactly.

Cite as

Mikkel Abrahamsen, Mark de Berg, Kevin Buchin, Mehran Mehr, and Ali D. Mehrabi. Range-Clustering Queries. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 5:1-5:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{abrahamsen_et_al:LIPIcs.SoCG.2017.5,
  author =	{Abrahamsen, Mikkel and de Berg, Mark and Buchin, Kevin and Mehr, Mehran and Mehrabi, Ali D.},
  title =	{{Range-Clustering Queries}},
  booktitle =	{33rd International Symposium on Computational Geometry (SoCG 2017)},
  pages =	{5:1--5:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-038-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{77},
  editor =	{Aronov, Boris and Katz, Matthew J.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2017.5},
  URN =		{urn:nbn:de:0030-drops-72147},
  doi =		{10.4230/LIPIcs.SoCG.2017.5},
  annote =	{Keywords: Geometric data structures, clustering, k-center problem}
}

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Documents authored by Mehrabi, Ali D.

Minimum Perimeter-Sum Partitions in the Plane

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Range-Clustering Queries

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