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Abstract

Recently there has been considerable research activity on algorithms for finding shortest paths in geometries induced by obstacles. A typical problem is finding the shortest path between two points on the Euclidean plane avoiding a given set of polygonal obstacles (see Figure 1). We review this area and isolate several interesting open problems.

Keywords

Short Path Short Path Problem Visibility Graph Polyhedral Surface Cylindrical Algebraic Decomposition 
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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References

  1. [AAGHI]
    T. Asano, T. Asano, L. J. Guibas, J. Hirshberger, H. Imai “Visibility Polygon Search in Euclidean Shortest Paths,” Proc. 26th FOCS Conference, 1985.Google Scholar
  2. [Baj]
    C. Bajaj “Applying Galois-Theoretic Algebraic Methods to Goemetric Optimization Problems,” Proc. SIAM Conference on Geometric Modeling and Robotics, 1985.Google Scholar
  3. [Bak]
    B. Baker “Shortest Paths with Unit Clearance Among Polygonal Obstacles,” Proc. SIAM Conference on Geometric Modeling and Robotics, 1985.Google Scholar
  4. [CR]
    J. Canny, J. H. Reif, private conversation, Loutraki Greece, 1986.Google Scholar
  5. [Ch]
    L. P. Chew “Planning the Shortest Path of a Disk in O(n 2 log n) Time,” Proc. First ACM Conference on Computational Geometry, 1985.Google Scholar
  6. [Co]
    G. E. Collins “Quantifier Elimination for Real Closed Fields by Cylindrical Algebraic Decomposition,” Proc. 2nd GI Conference on Automata Theory and Formal Languages, Springer-Verlag LNCS 35, 1975.Google Scholar
  7. [Di]
    E. W. Dijkstra “A Note on Two Problems in Connection With Graphs,” Numerische Mathematik, 1, pp. 269–271, 1959.Google Scholar
  8. [Ea]
    B. C. Eaves, private communication, Stanford University, 1985.Google Scholar
  9. [FT]
    M. Fredman, R. E. Tarjan “Fibonacci Heaps, and Their Uses in Improved Network Optimization Algorithms,” Proc. 25th FOCS Conference, 1985.Google Scholar
  10. [GJ]
    M. R. Garey, D. S. Johnson Computers and Intractability: A guide to the Theory of NP-completeness, Freeman, 1979Google Scholar
  11. [HSS]
    J. E. Hopcroft, J. T. Schwartz, M. Sharir “On the Complexity of Motion Planning for Multiple Independent Objects: PSPACE Hardness of the Warehouseman's Problem,” Robotics Research, 3, 4, pp. 76–88, 1984.Google Scholar
  12. [Jo]
    S. T. Jones “Solving Problems Involving Variable Terrain,” Byte, 5, 2, 1980.Google Scholar
  13. [LP]
    D. T. Lee, F. P. Preparata “Euclidean Shortest Paths in the Presence of Rectlinear Obstacles”, Networks, 14, pp. 393–410, 1984.Google Scholar
  14. [LS]
    D. Leven, M. Sharir “An Efficient and Simple Motion Planning Algorithm for a Ladder Moving in Two-dimensional Space Amidst Polygonal Barriers” Technical Report, Tel Aviv University, 1984.Google Scholar
  15. [LW]
    T. Lozano-Perez, M. A. Wesley “An Algorithm for Planning Collision-Free Paths Among Polyhedral Obstacles”, C.ACM, 22, 10, pp. 560–571, 1979.Google Scholar
  16. [Mi1]
    J. S. B. Mitchell “Shortest Paths in the Plane Among Obstacles,” Technical Report, Dept. of Operations Research, Stanford University, 1984.Google Scholar
  17. [Mi2]
    J. S. B. Mitchell “Shortest Rectlinear Paths Among Obstacles,” Technical Report, Dept. of Operations Research, Stanford University, 1986.Google Scholar
  18. [Mi3]
    J. S. B. Mitchell “Planning Shortest Paths,” Ph.D. Dissertation, Dept. of Operations Research, Stanford University, 1986.Google Scholar
  19. [MMP]
    J. S. B. Mitchell, D. M. Mount, C. H. Papadimitriou “The Discrete Geodesic Problem,” Technical Report, Dept. of Operations Research, Stanford University, 1985. To appear in SIAM J. Computing.Google Scholar
  20. [MP]
    J. S. B. Mitchell, C. H. Papadimitriou “The Weighted Regions Problem,” Technical Report, Dept. of Operations Research, Stanford University, 1986.Google Scholar
  21. [MZ]
    J. S. B. Mitchell, K. Zikan, private communication, Stanford University, 1986.Google Scholar
  22. [OSB]
    J. O'Rourke, S. Suri, H. Booth “Shortest Paths on Polyhedral Surfaces,” Technical Report, Johns Hopkins University, 1984.Google Scholar
  23. [Pa]
    C. H. Papadimitriou “An Algorithm for Shortest Path Motion in Three Dimensions”, Information Processing Letters, 20, pp. 249–263, 1985.Google Scholar
  24. [PSi]
    C. H. Papadimitriou, E. B. Silverberg “Optimal Piecewise-linear Motion of an Object Among Obstacles”, Technical Report, Dept. of Operations Research, Stanford University, 1986.Google Scholar
  25. [PS]
    C. H. Papadimitriou, K. Steiglitz Combinatorial Optimization: Algorithms and Complexity, Prentice-Hall, 1982Google Scholar
  26. [RS]
    J. H. Reif, J. A. Storer “Shortest Paths in Euclidean Space with Polyhedral Obstacles”, TR CS-85-121, Brandeis University, 1985.Google Scholar
  27. [ScS]
    J. T. Schwartz, M. Sharir “On the Piano Mover's Problem I,” International J. of Robotics Research, 2, 3, pp. 46–76, 1983.Google Scholar
  28. [ShS]
    M. Sharir, A. Schorr “On Shortest Paths in Polyhedral Spaces,” SIAM J. Computing, 15, 1, pp. 193–215, 1986.Google Scholar
  29. [Si]
    E. B. Silverberg, Ph.D. Thesis in preparation, Stanford University, 1986.Google Scholar
  30. [We]
    E. Welzl “Constructing the Visibility Graph for n Line Segments in O(n 2) time” Information Processing Letters, 20, pp. 167–171, 1985.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • Christos H. Papadimitriou
    • 1
    • 2
  1. 1.Stanford UniversityUSA
  2. 2.National Technical University of AthensGreece

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