, Volume 44, Issue 1, pp 1–19 | Cite as

Computing the external geodesic diameter of a simple polygon

  • D. Samuel
  • G. T. Toussaint


Given a simple polygonP ofn vertices, we present an algorithm that finds the pair of points on the boundary ofP that maximizes theexternal shortest path between them. This path is defined as theexternal geodesic diameter ofP. The algorithm takes0(n2) time and requires0(n) space. Surprisingly, this problem is quite different from that of computing theinternal geodesic diameter ofP. While the internal diameter is determined by a pair of vertices ofP, this is not the case for the external diameter. Finally, we show how this algorithm can be extended to solve theall external geodesic furthest neighbours problem.

AMS Subject Classifications

68U05 68C25 

Key words

Polygon geodesics diameter furthest neighbour algorithm complexity computational geometry 

Die Berechnung des äußeren geodätischen Durchmessers eines einfachen Vielecks


Gegeben sei ein einfaches PolygonP mitn Ecken. Wir geben einen Algorithmus an, der ein Punktepaar auf der Begrenzung vonP liefert, welches die Länge des kürzesten Weges maximiert, der im Äußeren des Polygons verläuft. Den Weg bezeichnen wir als den äußeren geodätischen Durchmesser vonP. Unser Algorithmus benötigt 0(n2) Zeit und erfordert 0(n) Speicherplatz. Zu unserer Überraschung ist das Problem von dem, der Berechnung des inneren geodätischen Durchmessers vonP völlig verschieden. Während der innere Durchmesser immer in Ecken vonP endet, muß dies für den äußeren Durchmesser nicht der Fall sein. Schließlich zeigen wir noch, daß der Algorithmus so erweitert werden kann, daß er das Problem der entferntesten äußeren geodätischen Nachbarn löst.


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  1. [1]
    T. Asano, L. Guibas, J. Hershberger, and H. Imai. Visibility-Polygon Search and Euclidean Shortest Paths. Proc. of 26th IEEE Symp. on Foundations of Computer Science, 1985, pp. 155–164.Google Scholar
  2. [2]
    T. Asano and G. Toussaint. Computing the Geodesic Center of a Simple Polygon. Discrete Algorithms and Complexity, Proc. of the Japan-U.S. Joint Seminar, June 4–6, 1986, Kyoto, Japan, pp. 65–79.Google Scholar
  3. [3]
    B. Chazelle. A Theorem on Polygon Cutting with Applications. Proc. of 23rd IEEE Symp. on Foundations of Computer Science, 1982, pp. 339–349.Google Scholar
  4. [4]
    H. Davenport and A. Schinzel. A Combinatorial Problem Connected with Differential Equations. Amer. J. Math.,87, 1965, pp. 684–694.Google Scholar
  5. [5]
    L. Guibas and J. Hershberger. Optimal Shortest Path Queries in a Simple Polygon. Proc. of the Third Ann. Symp. on Comp. Geom., 1987, pp. 64–75.Google Scholar
  6. [6]
    L. Guibas, J. Hershberger, D. Leven, M. Sharir, and R. E. Tarjan. Linear Time Algorithms for Visibility and Shortest Path Problems Inside Simple Polygons. Algorithmica,2, 1987, pp. 175–193.Google Scholar
  7. [7]
    L. Guibas, E. McCreight, M. Plass, and J. Roberts. A New Representation for Linear Lists. Proc. 9th ACM Symp. on Theory of Computing, 1977, pp. 49–60.Google Scholar
  8. [8]
    S. Huddleston and K. Mehlhorn. A New Data Structure for Representing Sorted Lists. Acta Informatica,17, 1982, pp. 157–184.Google Scholar
  9. [9]
    d. Kirkpatrick. Optimal Search in Planar Subdivisions. SIAM J. Computing,12, 1983, pp. 28–35.Google Scholar
  10. [10]
    D. T. Lee and F. P. Preparata. Euclidean Shortest Paths in the Presence of Rectilinear Barriers. Networks, vol. 14, 3, 1984, pp. 393–410.Google Scholar
  11. [11]
    D. Mount. On Finding Shortest Paths on Convex Polyhedra. Tech. Rept., University of Maryland, May 1985.Google Scholar
  12. [12]
    R. Pollack, G. Rote and M. Sharir. Computing the Geodesic Center of a Simple Polygon. Tech. Rept. 231, Computer Science Department, Courant Institute, July 1986.Google Scholar
  13. [13]
    R. Pollack and M. Sharir. Computing the Geodesic Center of a Simple Polygon in “Research Workshop on Moveable Separability of Sets,” G. Toussaint, ed., Bellairs Research Institute of McGill University, February 1986, pp. 26–47.Google Scholar
  14. [14]
    J. Reif and J. Storer. Shortest Paths in Euclidean Space with Polyhedral Obstacles. Tech. Rept., Brandeis University, April 1985.Google Scholar
  15. [15]
    J. Reif and J. Storer. Minimizing Turns for Discrete Movement in the Interior of a Polygon. Tech. Rept., Harvard University, December 1985.Google Scholar
  16. [16]
    M. I. Shamos. Problems in Computational Geometry. PhD thesis, Carnegie-Mellon University, 1977.Google Scholar
  17. [17]
    M. Sharir and A. Schorr. On Shortest Paths in Polyhedral Spaces. Proc. 16th ACM Symp. on Theory of Computing, 1984, pp. 144–153.Google Scholar
  18. [18]
    S. Suri. The All-Geodesic-Furthest Neighbours Problem for Simple Polygons. Proc. of the Third Ann. Symp. on Comp. Geom., 1987, pp. 64–75.Google Scholar
  19. [19]
    E. Szemeredi. On a Problem by Davenport and Schinzel. Acta Arith.,25, 1974, pp. 213–224.Google Scholar
  20. [20]
    G. Toussaint. Computing Geodesic Properties Inside a Simple Polygon. Revue d'Intelligence Artificielle, 3, 1989, pp. 9–42.Google Scholar

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • D. Samuel
    • 1
  • G. T. Toussaint
    • 1
  1. 1.School of Computer ScienceMcGill UniversityMontréalCanada

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