Skip to main content
SpringerLink
Log in

On bends and lengths of rectilinear paths: A graph-theoretic approach

  • Conference paper
  • First Online:
Algorithms and Data Structures (WADS 1991)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 519))

Included in the following conference series:

Abstract

We consider the problem of finding a rectilinear path between two designated points in the presence of rectilinear obstacles subject to various optimization functions in terms of the number of bends and the total length of the path. Specifically we are interested in finding a minimum bend shortest path, a shortest minimum bend path or a least-cost path where the cost is defined as a function of both the length and the number of bends of the path. We provide a unified approach by constructing a path-preserving graph guaranteed to preserve all these three kinds of paths and give an O(K+e log e) algorithm to find them, where e is the total number of obstacle edges, and K is the number of intersections between tracks from extreme point and other tracks (defined in the text). K is bounded by O(et), where t is the number of extreme edges. In particular, if the obstacles are rectilinearly convex, then K is O(ne), where n is the number of obstacles. Extensions are made to find a shortest path with a bounded number of bends and a minimum-bend path with a bounded length. When a source point and obstacles are pre-given, queries for the assorted paths from the source to given points can be handled in O(log n+k) time after O(K+e log e) preprocessing, where k is the size of the goal path. The trans-dichotomous algorithm of Fredman and Willard[8] and the running time for these problems are also discussed.

Supported in part by the National Science Foundation under Grant CCR-8901815.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. T. Asano, T. Asano, L. Guibas, J. Hershberger, and H. Imai. Visibility of disjoint polygons. Algorithmica, 1, 1986, pp.49–63.

    Google Scholar 

  2. B. Chazelle. Triangulating a simple polygon in linear time. Tech. Report CS-TR-264-90, Princeton Univ., 1990.

    Google Scholar 

  3. K. L. Clarkson, S. Kapoor, and P. M. Vaidya. Rectilinear shortest paths through polygonal obstacles in O(n log3/2 n) time. Submitted for publication.

    Google Scholar 

  4. M. de Berg, M. van Kreveld, B. J. Nillson, M. H. Overmars. Finding shortest paths in the presence of orthogonal obstacles using a combined L 1 and link metric. Proc. of SWAT '90, Lect. Notes in Computer Science, 447, Springer-Verlag, 1990, pp.213–224.

    Google Scholar 

  5. P. J. deRezende, D. T. Lee, and Y. F. Wu. Rectilinear shortest paths with rectangular barriers. Discrete and Computational Geometry, 4, 1989, pp. 41–53.

    Google Scholar 

  6. H. N. Djidjev, A. Lingas and J. Sack. An O(n log n) algorithm for computing the link center in a simple polygon. Proc. of STACS '89, Lect. Notes in Computer Science, 349, Springer-Verlag, 1989, pp.96–107.

    Google Scholar 

  7. M. L. Fredman and R. E. Tarjan. Fibonacci heaps and their uses in improved network optimization algorithms. Proc. 25th IEEE Sympo. on Foundations of Computer Science, 1984, pp.338–346.

    Google Scholar 

  8. M. L. Fredman and D. E. Willard. Trans-dichotomous algorithms for Minimum Spanning Trees and Shortest Paths. Proc. 31th IEEE Sympo. on Foundations of Computer Science, 1990, pp.719–725.

    Google Scholar 

  9. M. R. Garey and D. S. Johnson. Computers and Intractability. A Guide to the Theory of NP-Completeness. W. H. Freeman, San Francisco, 1979, pp.214.

    Google Scholar 

  10. L. J. Guibas and J. Hershberger. Optimal shortest path queries in a simple polygon. Proc 3rd ACM Symp. on Computational Geometry, 1987, pp.50–63.

    Google Scholar 

  11. H. Imai and T. Asano. Dynamic Segment Intersection Search with Applications. Proc. 25th IEEE Symp. on Foundations of Computer Science, Springer Island, Florida, 1984, pp.393–402.

    Google Scholar 

  12. Y. Ke. An Efficient Algorithm for Link-Distance Problems. Proc. 5th ACM Symp. on Computational Geometry, 1989, pp.69–78.

    Google Scholar 

  13. R. C. Larson and V. O. Li. Finding minimum rectilinear distance paths in the presence of barriers. Networks, 11, 1981, pp.285–304.

    Google Scholar 

  14. D. T. Lee. Proximity and reachability in the plane. PhD. Dissertation, University of Illinois, 1978.

    Google Scholar 

  15. D. T. Lee and F. P. Preparata. Euclidean shortest paths in the presence of rectilinear barriers. Networks, 14, 1984, pp.393–410.

    Google Scholar 

  16. Lenhart, R. Pollack, J. Sack, R. Seidel, M. Sharir, S. Suri, G. Toussaint, S. Whitesides and C. Yap. Computing the link center of a simple polygon. Proc. 3rd ACM Symp. on Computational Geometry, 1987, pp.1–10

    Google Scholar 

  17. T. Lozano-Perez and M. A. Wesley. An algorithm for planning collision-free paths among polyhedral obstacles. CACM, 22, 1979, pp.560–570.

    Google Scholar 

  18. H. G. Mairson and J. Stolfi. Reporting line segment intersections in the plane. Tech. Rep., Dept. of Computer Sci., Stanford University, 1983.

    Google Scholar 

  19. K. M. McDonald and J. G. Peters. Smallest paths in simple rectilinear polygons. Submitted for publication.

    Google Scholar 

  20. J. S. B. Mitchell and C. Papadimitriou. Planning shortest paths. SIAM conference July 15–19, 1985.

    Google Scholar 

  21. J. S. B. Mitchell. Shortest rectilinear paths among obstacles. Technical report NO. 739, School of OR/IE, Cornell University, April 1987.

    Google Scholar 

  22. J. S. B. Mitchell. An optimal algorithm for shortest rectilinear paths among obstacles in the plane. Abstracts of the First Canadian Conference on Computational Geometry, 1989, pp.22.

    Google Scholar 

  23. J. S. B. Mitchell, G. Rote and G. Wōginger. Computing the Minimum Link Path among a Set of Obstacles in the Planes. Proc. 6th ACM Symp. on Computational Geometry, 1990.

    Google Scholar 

  24. T. Ohtsuki. Gridless Routers — New Wire Routing Algorithm Based on Computational Geometry. International Conference on Circuits and Systems, China, 1985.

    Google Scholar 

  25. S. Suri. A Linear Time Algorithm for Minimum Link Paths Inside a Simple Polygon. Computer Vision, Graphics and Image Processing 35, 1986, pp.99–110.

    Google Scholar 

  26. E. Welzl. Constructing the visibility graph for n line segments in O(n 2) time. Info. Proc. Lett., 1985, pp.167–171.

    Google Scholar 

  27. P. Widmayer. Network design issues in VLSI. Manuscript, 1989.

    Google Scholar 

  28. Y. F. Wu, P. Widmayer, M. D. F. Schlag, and C. K. Wong. Rectilinear shortest paths and minimum spanning trees in the presence of rectilinear obstacles. IEEE Trans. Comput., 1987, pp.321–331.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of EE & CS, Northwestern University, 60208, Evanston, IL

    C. D. Yang & D. T. Lee

  2. IBM Research Division, Thomas J. Watson Research Center, 10598, Yorktown Heights, NY

    C. K. Wong

Authors
  1. C. D. Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. T. Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. K. Wong
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Frank Dehne Jörg-Rüdiger Sack Nicola Santoro

Rights and permissions

Reprints and permissions

Copyright information

© 1991 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Yang, C.D., Lee, D.T., Wong, C.K. (1991). On bends and lengths of rectilinear paths: A graph-theoretic approach. In: Dehne, F., Sack, JR., Santoro, N. (eds) Algorithms and Data Structures. WADS 1991. Lecture Notes in Computer Science, vol 519. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0028272

Download citation

  • DOI: https://doi.org/10.1007/BFb0028272

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-54343-5

  • Online ISBN: 978-3-540-47566-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation