Computing minimum length paths of a given homotopy class

  • John Hershberger
  • Jack Snoeyink
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 519)


In this abstract, we use the universal covering space of a surface to generalize previous results on computing paths in a simple polygon. We look at optimizing paths among obstacles in the plane under the Euclidean and link metrics and polygonal convex distance functions. The universal cover is a unifying framework that reveals connections between minimum paths under these three distance functions, as well as yielding simpler linear-time algorithms for shortest path trees and minimum link paths in simple polygons.


Short Path Homotopy Class Simple Polygon Short Path Tree Universal Covering Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    A. Aggarwal, H. Booth, J. O'Rourke, S. Suri, and C. K. Yap. Finding minimal convex nested polygons. Information and Computation, 83(1):98–110, Oct. 1989.Google Scholar
  2. [2]
    M. A. Armstrong. Basic Topology. McGraw-Hill, London, 1979.Google Scholar
  3. [3]
    B. Baumgart. A polyhedral representation for computer vision. In Proceedings of the AFIPS National Computer Conference, pages 589–596, 1975.Google Scholar
  4. [4]
    B. Bhattacharya and G. T. Toussaint. A linear algorithm for determining translation separability of two simple polygons. Technical Report SOCS-86.1, School of Computer Science, McGill University, Montreal, 1986.Google Scholar
  5. [5]
    J. W. S. Cassels. An Introduction to the Geometry of Numbers. Springer-Verlag, Berlin, 1959.Google Scholar
  6. [6]
    B. Chazelle. A theorem on polygon cutting with applications. In Proceedings of the 23rd IEEE Symposium on Foundations of Computer Science, pages 339–349, 1982.Google Scholar
  7. [7]
    B. Chazelle. Triangulating a simple polygon in linear time. In Proceedings of the 31st IEEE Symposium on Foundations of Computer Science, pages 220–230, 1990.Google Scholar
  8. [8]
    K. L. Clarkson, S. Kapoor, and P. M. Vaidya. Rectilinear shortest paths through polygonal obstacles in O(n log2 n) time. In Proceedings of the Third Annual ACM Symposium on Computational Geometry, pages 251–257, 1987.Google Scholar
  9. [9]
    J. Czyzowicz, P. Egyed, H. Everett, D. Rappaport, T. Shermer, D. Souvaine, G. Toussaint, and J. Urrutia. The aquarium keeper's problem. In Proceedings of the Second Annual ACM-SIAM Symposium on Discrete Algorithms, pages 459–464, Jan. 1991.Google Scholar
  10. [10]
    M. de Berg. On rectilinear link distance. Technical Report RUU-CS-89-13, Vakgroep Informatica, Rijksuniversiteit Utrecht, May 1989.Google Scholar
  11. [11]
    M. de Berg. Translating polygons with applications to hidden surface removal. In SWAT 90: Second Scandinavian Workshop on Algorithm Theory, number 447 in Lecture Notes in Computer Science, pages 60–70. Springer-Verlag, 1990.Google Scholar
  12. [12]
    P. J. de Rezende, D. T. Lee, and Y. F. Wu. Rectilinear shortest paths with rectangular barriers. Discrete and Computational Geometry, 4:41–53, 1989.Google Scholar
  13. [13]
    S. Gao, M. Jerrum, M. Kaufmann, K. Mehlhorn, W. Rülling, and C. Storb. On continuous homotopic one layer routing. In Proceedings of the Third Annual ACM Symposium on Computational Geometry, pages 392–402, 1987.Google Scholar
  14. [14]
    S. K. Ghosh. Computing the visibility polygon from a convex set and related problems. Journal of Algorithms, 12:75–95, 1991.Google Scholar
  15. [15]
    S. K. Ghosh and D. M. Mount. An output sensitive algorithm for computing visibility graphs. In Proceedings of the 28th IEEE Symposium on Foundations of Computer Science, pages 11–19, 1987.Google Scholar
  16. [16]
    R. Graham. An efficient algorithm for determining the convex hull of a finite planar set. Information Processing Letters, 1:132–133, 1972.Google Scholar
  17. [17]
    L. Guibas, J. Hershberger, D. Leven, M. Sharir, and R. Tarjan. Linear time algorithms for visibility and shortest path problems inside triangulated simple polygons. Algorithmica, 2:209–233, 1987.Google Scholar
  18. [18]
    L. Guibas and J. Stolfi. Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams. ACM Transactions on Graphics, 4(2):74–123, 1985.Google Scholar
  19. [19]
    L. J. Guibas and J. Hershberger. Optimal shortest path queries in a simple polygon. Journal of Computer and System Sciences, 39(2):126–152, Oct. 1989.Google Scholar
  20. [20]
    R. H. Güting. Conquering Contours: Efficient Algorithms for Computational Geometry. PhD thesis, Dortmund, 1983.Google Scholar
  21. [21]
    R. H. Gütting and T. Ottmann. New algorithms for special cases of hidden line elimination problem. Computer Vision, Graphics, and Image Processing, 40:188–204, 1987.Google Scholar
  22. [22]
    S. Kapoor and S. N. Maheshwari. Efficient algorithms for Euclidean shortest path and visibility problems with polygonal obstacles. In Proceedings of the Fourth Annual ACM Symposium on Computational Geometry, pages 172–182, 1988.Google Scholar
  23. [23]
    R. C. Larson and V. O. Li. Finding minimum rectilinear paths in the presence of barriers. Networks, 11:285–304, 1981.Google Scholar
  24. [24]
    D. T. Lee and F. P. Preparata. Euclidean shortest paths in the presence of rectilinear barriers. Networks, 14(3):393–410, 1984.Google Scholar
  25. [25]
    C. E. Leiserson and F. M. Maley. Algorithms for routing and testing routability of planar VLSI layouts. In Proceedings of the 17th Annual ACM Symposium on Theory of Computing, pages 69–78, 1985.Google Scholar
  26. [26]
    J. R. Munkres. Topology: A First Course. Prentice-Hall, Englewood Cliffs, N.J., 1975.Google Scholar
  27. [27]
    F. P. Preparata and M. I. Shamos. Computational Geometry—An Introduction. Springer-Verlag, New York, 1985.Google Scholar
  28. [28]
    G. J. E. Rawlins and D. Wood. Optimal computation of finitely oriented convex hulls. Information and Computation, 72:150–166, 1987.Google Scholar
  29. [29]
    J.-R. Sack. Rectilinear Computational Geometry. PhD thesis, Carleton University, 1984.Google Scholar
  30. [30]
    S. Suri. A linear time algorithm for minimum link paths inside a simple polygon. Computer Vision, Graphics, and Image Processing, 35:99–110, 1986.Google Scholar
  31. [31]
    X.-H. Tan, T. Hirata, and Y. Inagaki. The intersection searching problem for c-oriented polygons. Information Processing Letters, 37:201–204, 1991.Google Scholar
  32. [32]
    G. Toussaint. On separating two simple polygons by a single translation. Discrete and Computational Geometry, 4:265–278, 1989.Google Scholar
  33. [33]
    G. T. Toussaint. Movable separability of sets. In G. T. Toussaint, editor, Computational Geometry, volume 2 of Machine Intelligence and Pattern Recognition, pages 335–376. North Holland, Amsterdam, 1985.Google Scholar
  34. [34]
    G. T. Toussaint. Computing geodesic properties inside a simple polygon. Revue D'Intelligence Artificielle, 3(2):9–42, 1989. Also available as technical report SOCS 88.20, School of Computer Science, McGill University.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • John Hershberger
    • 1
  • Jack Snoeyink
    • 2
  1. 1.DEC Systems Research CenterUSA
  2. 2.Department of Computer ScienceUtrecht UniversityThe Netherlands

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