Advertisement

Minimum Perimeter-Sum Partitions in the Plane

  • Mikkel AbrahamsenEmail author
  • Mark de Berg
  • Kevin Buchin
  • Mehran Mehr
  • Ali D. Mehrabi
Article
  • 11 Downloads

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 \({\textsc {ch}}(P_1)\) and \({\textsc {ch}}(P_2)\) is minimized, where \({\textsc {ch}}(P_i)\) 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 \(O(n \log ^2 n)\) time and a \((1+\varepsilon )\)-approximation algorithm running in \(O(n + 1/\varepsilon ^2\cdot \log ^2(1/\varepsilon ))\) time.

Keywords

Computational geometry Clustering Minimum-perimeter partition Convex hull 

Mathematics Subject Classification

68W05 

Notes

Acknowledgements

This research was initiated when the first author visited the Department of Computer Science at TU Eindhoven during the winter 2015–2016. He wishes to express his gratitude to the other authors and the department for their hospitality.

Funding

M. Abrahamsen is supported by the Advanced Grant DFF-0602-02499B from the Danish Council for Independent Research under the Sapere Aude research career programme. M. de Berg, K. Buchin, M. Mehr, and A. D. Mehrabi are supported by the Netherlands’ Organisation for Scientific Research (NWO) under Project No. 024.002.003, 612.001.207, 022.005025, and 612.001.118 respectively.

References

  1. 1.
    Abrahamsen, M., Adamaszek, A., Bringmann, K., Cohen-Addad, V., Mehr, M., Rotenberg, E., Roytman, A., Thorup, M.: Fast fencing. In: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing (STOC’18), pp. 564–573. ACM, New York (2018)Google Scholar
  2. 2.
    Abrahamsen, M., de Berg, M., Buchin, K., Mehr, M., Mehrabi, A.D.: Minimum perimeter-sum partitions in the plane. In: Aronov, B., Katz, M.J. (eds.) Proceedings of the 33rd International Symposium on Computational Geometry (SoCG’17), Article No. 4. LIPIcs. Leibniz International Proceedings in Informatics, vol. 77. Schloss Dagstuhl, Leibniz-Zentrum für Informatik, Wadern (2017)Google Scholar
  3. 3.
    Agarwal, P.K., Sharir, M.: Efficient algorithms for geometric optimization. ACM Comput. Surv. 30(4), 412–458 (1998)CrossRefGoogle Scholar
  4. 4.
    Arkin, E.M., Khuller, S., Mitchell, J.S.B.: Geometric knapsack problems. Algorithmica 10(5), 399–427 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Asano, T., Bhattacharya, B., Keil, M., Yao, F.: Clustering algorithms based on minimum and maximum spanning trees. In: Proceedings of the 4th Annual Symposium on Computational Geometry (SoCG’88), pp. 252–257. ACM, New York (1988)Google Scholar
  6. 6.
    Bae, S.W., Cho, H.-G., Evans, W., Saeedi, N., Shin, C.-S.: Covering points with convex sets of minimum size. Theor. Comput. Sci. 718, 14–23 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Capoyleas, V., Rote, G., Woeginger, G.: Geometric clusterings. J. Algorithms 12(2), 341–356 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chan, T.M.: More planar two-center algorithms. Comput. Geom. 13(2), 189–198 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer, Berlin (2008)CrossRefzbMATHGoogle Scholar
  10. 10.
    Devillers, O., Katz, M.J.: Optimal line bipartitions of point sets. Int. J. Comput. Geom. Appl. 9(1), 39–51 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Drezner, Z.: The planar two-center and two-median problems. Transp. Sci. 18(4), 351–361 (1984)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Eppstein, D.: Faster construction of planar two-centers. In: Proceedings of the 8th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’17), pp. 131–138. ACM, New York (1997)Google Scholar
  13. 13.
    Har-Peled, S.: Geometric Approximation Algorithms. Mathematical Surveys and Monographs, vol. 173. American Mathematical Society, Providence (2011)Google Scholar
  14. 14.
    Hershberger, J.: Minimizing the sum of diameters efficiently. Comput. Geom. 2(2), 111–118 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Jaromczyk, J.W., Kowaluk, M.: An efficient algorithm for the Euclidean two-center problem. In: Proceedings of the 10th ACM Symposium on Computational Geometry (SoCG’94), pp. 303–311. ACM, New York (1994)Google Scholar
  16. 16.
    Lee, D.T., Wu, Y.F.: Geometric complexity of some location problems. Algorithmica 1(2), 193–211 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mitchell, J.S.B., Wynters, E.L.: Finding optimal bipartitions of points and polygons. In: Dehne, F., et al. (eds.) Algorithms and Data Structures (WADS’91). Lecture Notes in Computer Science, vol. 519, pp. 202–213. Springer, Berlin (1991). http://www.ams.sunysb.edu/~jsbm/
  18. 18.
    Oh, E., Ahn, H.-K.: Polygon queries for convex hulls of points. In: Wang, L., Zhu, D. (eds.) Computing and Combinatorics (COCOON’18). Lecture Notes in Computer Science, vol. 10976, pp. 143–155. Springer, Cham (2018)CrossRefGoogle Scholar
  19. 19.
    Rokne, J., Wang, S., Wu, X.: Optimal bipartitions of point sets. In: Proceedings of the 4th Canadian Conference on Computational Geometry (CCCG’92), pp. 11–16. Memorial University of Newfoundland, St. John’s (1992)Google Scholar
  20. 20.
    Segal, M.: Lower bounds for covering problems. J. Math. Model. Algorithms 1(1), 17–29 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Sharir, M.: A near-linear algorithm for the planar 2-center problem. Discrete Comput. Geom. 18(2), 125–134 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of CopenhagenCopenhagenDenmark
  2. 2.Department of Computer ScienceTU EindhovenEindhovenThe Netherlands

Personalised recommendations