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Algorithmica

, Volume 8, Issue 1–6, pp 55–88 | Cite as

L 1 shortest paths among polygonal obstacles in the plane

  • Joseph S. B. Mitchell
Article

Abstract

We present an algorithm for computingL 1 shortest paths among polygonal obstacles in the plane. Our algorithm employs the “continuous Dijkstra” technique of propagating a “wavefront” and runs in timeO(E logn) and spaceO(E), wheren is the number of vertices of the obstacles andE is the number of “events.” By using bounds on the density of certain sparse binary matrices, we show thatE =O(n logn), implying that our algorithm is nearly optimal. We conjecture thatE =O(n), which would imply our algorithm to be optimal. Previous bounds for our problem were quadratic in time and space.

Our algorithm generalizes to the case of fixed orientation metrics, yielding anO(nɛ−1/2 log2 n) time andO(nɛ−1/2) space approximation algorithm for finding Euclidean shortest paths among obstacles. The algorithm further generalizes to the case of many sources, allowing us to compute anL 1 Voronoi diagram for source points that lie among a collection of polygonal obstacles.

Key words

Shortest paths Voronoi diagrams Rectilinear paths Wire routing Fixed orientation metrics Continuous Dijkstra algorithm Computational geometry Extremal graph theory 

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References

  1. [As]
    T. Asano, Rectilinear shortest paths in a rectilinear simple polygon,Trans. IECE of Japan,E69 (1985), 750–758.Google Scholar
  2. [AAGHI]
    T. Asano, T. Asano, L. Guibas, J. Hershberger, and H. Imai, Visibility of disjoint polygons,Algorithmica,1 (1986), 49–63.MATHCrossRefMathSciNetGoogle Scholar
  3. [BG]
    D. Bienstock and E. Györi, An extremal problem on sparse 0–1 matrices,SIAM Journal on Discrete Mathematics,4 (1991), 17–27.MATHCrossRefMathSciNetGoogle Scholar
  4. [Bo]
    B. Bollobás,Extremal Graph Theory, Academic Press, New York, 1978.MATHGoogle Scholar
  5. [Ch1]
    B. Chazelle, A functional approach to data structures and its use in multidimensional searching,SIAM Journal on Computing,17 (1988). 427–462.MATHCrossRefMathSciNetGoogle Scholar
  6. [Ch2]
    B. Chazelle, An algorithm for segment dragging and its implementation,Algorithmica,3 (1988), 205–221.CrossRefMathSciNetGoogle Scholar
  7. [CEGS]
    B. Chazelle, H. Edelsbrunner, L. Guibas, and M. Sharir, Lines in space—Combinatorics, algorithms and applications,Proc. 21st Annual ACM Symposium on Theory of Computing, 1989, pp. 382–393.Google Scholar
  8. [Cl]
    K. Clarkson, Approximation algorithms for shortest path motion planning,Proc. 19th Annual ACM Symposium on Theory of Computing, New York City, May 25–27, 1987, pp. 56–65.Google Scholar
  9. [CKV]
    K. Clarkson, S. Kapoor, and P. Vaidya, Rectilinear shortest paths through polygonal obstacles inO(n log2 n) time,Proc. Third Annual ACM Symposium on Computational Geometry; Waterloo, Ontario, 1987, pp. 251–257.Google Scholar
  10. [DLW]
    P. J. de Rezende, D. T. Lee, and Y. F. Wu, Rectilinear shortest paths with rectangular barriers,Proc. First Annual ACM Symposium on Computational Geometry, 1985, pp. 204–213.Google Scholar
  11. [Di]
    E. W. Dijkstra, A note on two problems in connexion with graphs,Numerische Mathematik,1 (1959), 269–271.MATHCrossRefMathSciNetGoogle Scholar
  12. [EGS]
    H. Edelsbrunner, L. J. Guibas, and J. Stolfi, Optimal point location in a monotone subdivision,SIAM Journal on Computing,16 (1986), 317–340.CrossRefMathSciNetGoogle Scholar
  13. [EOS]
    H. Edelsbrunner, M. H. Overmars, and R. Seidel, Some methods of computational geometry applied to computer graphics,Computer Vision, Graphics, and Image Processing,28 (1984), 92–108.CrossRefGoogle Scholar
  14. [Fo]
    S. J. Fortune, A sweepline algorithm for Voronoi diagrams,Algorithmica,2 (1987), 153–174.MATHCrossRefMathSciNetGoogle Scholar
  15. [FH]
    Z. Füredi and P. Hajnal, Davenport-Schinzel theory of matrices, Manuscript, Department of Mathematics, MIT, October, 1989.Google Scholar
  16. [KM]
    S. Kapoor and S. N. Maheshwari, Efficient algorithms for Euclidean shortest path and visibility problems with polygonal obstacles,Proc. Fourth Annual ACM Symposium on Computational Geometry, Urbana-Champaign, IL, June 6–8, 1988, pp. 172–182.Google Scholar
  17. [Ki]
    D. G. Kirkpatrick, Optimal search in planar subdivisions,SIAM Journal of Computing,12 (1983), 28–35.MATHCrossRefMathSciNetGoogle Scholar
  18. [LL]
    R. C. Larson and V. O. Li, Finding minimum rectilinear distance paths in the presence of barriers,Networks,11 (1981), 285–304.MATHCrossRefMathSciNetGoogle Scholar
  19. [Le]
    D. T. Lee, Proximity and reachability in the plane, Ph.D. Thesis, Technical Report R-831, Dept. of Electrical Engineering, University of Illinois, November 1978.Google Scholar
  20. [LCY]
    D. T. Lee, T. H. Chen, and C. D. Yang, Shortest rectilinear paths among weighted obstacles,Proc. Sixth Annual ACM Symposium on Computational Geometry, 1990, pp. 301–310.Google Scholar
  21. [LP]
    D. T. Lee and F. P. Preparata, Euclidean shortest paths in the presence of rectilinear boundaries,Networks,14 (1984), 393–410.MATHCrossRefMathSciNetGoogle Scholar
  22. [LWo]
    D. T. Lee and C. K. Wong, Voronoi diagrams inL 1-(L ) metrics with 2-dimensional storage applications,SIAM Journal of Computing,9(1) (1980), 200–211.MATHCrossRefMathSciNetGoogle Scholar
  23. [Lo]
    L. Lovász,Combinatorial Problems and Exercises, North-Holland, Amsterdam, 1979.MATHGoogle Scholar
  24. [LW]
    T. Lozano-Perez and M. A. Wesley, An algorithm for planning collision-free paths among polyhedral obstacles,Communications of the ACM,22 (1979), 560–570.CrossRefGoogle Scholar
  25. [Mi1]
    J. S. B. Mitchell, Planning shortest paths, Ph.D. Thesis, Department of Operations Research, Stanford University, August, 1986.Google Scholar
  26. [Mi2]
    J. S. B. Mitchell, Shortest rectilinear paths among obstacles, Technical Report No. 739, School of Operations Research and Industrial Engineering, Cornell University, April, 1987.Google Scholar
  27. [Mi3]
    J. S. B. Mitchell, A new algorithm for shortest paths among obstacles in the plane,Annals of Mathematics and Artificial Intelligence,3 (1991), 83–106.MATHCrossRefMathSciNetGoogle Scholar
  28. [MMP]
    J. S. B. Mitchell, D. M. Mount, and C. H. Papadimitriou, The discrete geodesic problem,SIAM Journal on Computing,16(4) (1987), 647–668.MATHCrossRefMathSciNetGoogle Scholar
  29. [MP]
    J. S. B. Mitchell and C. H. Papadimitriou, The weighted region problem: Finding shortest paths through a weighted planar subdivision,Journal of the ACM,38(1) (1991), 18–73.MATHCrossRefMathSciNetGoogle Scholar
  30. [Ni]
    N. J. Nilsson,Principles of Artifical Intelligence, Tioga Publishing Co., Palo Alto, CA, 1980.Google Scholar
  31. [PaS]
    J. Pach and M. Sharir, On vertical visibility in arrangements of segments and the queue size in the Bentley-Ottman line sweeping algorithm,First Canadian Conference on Computational Geometry, Montreal, Quebec, August 21–25, 1989. To appear:SIAM Journal on Computing.Google Scholar
  32. [Pa]
    C. H. Papadimitriou, An algorithm for shortest-path motion in three dimensions,Information Processing Letters,20 (1985), 259–263.MATHCrossRefMathSciNetGoogle Scholar
  33. [Pr]
    F. P. Preparata, A new approach to planar point location,SIAM Journal on Computing,10 (1981), 473–482.MATHCrossRefMathSciNetGoogle Scholar
  34. [PrS]
    F. P. Preparata and M. I. Shamos,Computational Geometry, Springer-Verlag, New York, 1985.Google Scholar
  35. [RS]
    J. H. Reif and J. A. Storer, Shortest paths in Euclidean space with polyhedral obstacles, Technical Report CS-85-121, Computer Science Department, Brandeis University, April, 1985.Google Scholar
  36. [Sch]
    E. Schmeichel, Private Communication, Department of Mathematics and Computer Science, San Jose State University, 1986.Google Scholar
  37. [Se]
    R. Seidel, Constrained Delaunay triangulations and Voronoi diagrams with obstacles, Technical Report Rep. 260, Institute für Information Processing, Graz, pp. 178–191, June, 1988.Google Scholar
  38. [Sh]
    M. Sharir,Davenport-Schinzel Sequences and their Geometric Applications, NATO ASI Series, Vol. F40, Theoretical Foundations of Computer Graphics and CAD (R. A. Earnshaw, ed.), Springer-Verlag, Berlin, 1988, pp. 253–278.Google Scholar
  39. [SCKLPS]
    M. Sharir, R. Cole, K. Kedem, D. Leven, R. Pollack, and S. Sifrony, Geometric applications of Davenport-Schinzel sequences,Proc. 27th Annual IEEE Symposium on Foundations of Computer Science, 1986, pp. 77–86.Google Scholar
  40. [SS]
    M. Sharir and A. Schorr, On shortest paths in polyhedral spaces,SIAM Journal on Computing,15 (1) (1986), 193–215.MATHCrossRefMathSciNetGoogle Scholar
  41. [We]
    E. Welzl, Constructing the visibility graph forn line segments inO(n 2) time,Information Processing Letters,20 (1985), 167–171.MATHCrossRefMathSciNetGoogle Scholar
  42. [Wi]
    P. Widmayer, Network design issues in VLSI, Manuscript, Institut für Informatik, University Freiburg, Rheinstra\e 10-12, 7800 Freiburg, West Germany, 1989.Google Scholar
  43. [WWW]
    P. Widmayer, Y. F. Wu, and C. K. Wong, On some distance problems in fixed orientations,SIAM Journal on Computing,16 (4), (1987), 728–746.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1992

Authors and Affiliations

  • Joseph S. B. Mitchell
    • 1
  1. 1.Department of Applied Mathematics and StatisticsState University of New York at Stony BrookStony BrookUSA

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