# Shortest-path motion

Invited Talk

First Online:

## 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

- [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 - [Baj]C. Bajaj “Applying Galois-Theoretic Algebraic Methods to Goemetric Optimization Problems,”
*Proc. SIAM Conference on Geometric Modeling and Robotics,*1985.Google Scholar - [Bak]B. Baker “Shortest Paths with Unit Clearance Among Polygonal Obstacles,”
*Proc. SIAM Conference on Geometric Modeling and Robotics,*1985.Google Scholar - [CR]J. Canny, J. H. Reif, private conversation, Loutraki Greece, 1986.Google Scholar
- [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 - [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 - [Di]E. W. Dijkstra “A Note on Two Problems in Connection With Graphs,”
*Numerische Mathematik,*1, pp. 269–271, 1959.Google Scholar - [Ea]B. C. Eaves, private communication, Stanford University, 1985.Google Scholar
- [FT]M. Fredman, R. E. Tarjan “Fibonacci Heaps, and Their Uses in Improved Network Optimization Algorithms,”
*Proc. 25th FOCS Conference,*1985.Google Scholar - [GJ]M. R. Garey, D. S. Johnson
*Computers and Intractability: A guide to the Theory of NP-completeness,*Freeman, 1979Google Scholar - [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 - [Jo]S. T. Jones “Solving Problems Involving Variable Terrain,”
*Byte, 5,*2, 1980.Google Scholar - [LP]D. T. Lee, F. P. Preparata “Euclidean Shortest Paths in the Presence of Rectlinear Obstacles”,
*Networks, 14,*pp. 393–410, 1984.Google Scholar - [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
- [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 - [Mi1]J. S. B. Mitchell “Shortest Paths in the Plane Among Obstacles,” Technical Report, Dept. of Operations Research, Stanford University, 1984.Google Scholar
- [Mi2]J. S. B. Mitchell “Shortest Rectlinear Paths Among Obstacles,” Technical Report, Dept. of Operations Research, Stanford University, 1986.Google Scholar
- [Mi3]J. S. B. Mitchell “Planning Shortest Paths,” Ph.D. Dissertation, Dept. of Operations Research, Stanford University, 1986.Google Scholar
- [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 - [MP]J. S. B. Mitchell, C. H. Papadimitriou “The Weighted Regions Problem,” Technical Report, Dept. of Operations Research, Stanford University, 1986.Google Scholar
- [MZ]J. S. B. Mitchell, K. Zikan, private communication, Stanford University, 1986.Google Scholar
- [OSB]J. O'Rourke, S. Suri, H. Booth “Shortest Paths on Polyhedral Surfaces,” Technical Report, Johns Hopkins University, 1984.Google Scholar
- [Pa]C. H. Papadimitriou “An Algorithm for Shortest Path Motion in Three Dimensions”,
*Information Processing Letters, 20,*pp. 249–263, 1985.Google Scholar - [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
- [PS]C. H. Papadimitriou, K. Steiglitz
*Combinatorial Optimization: Algorithms and Complexity,*Prentice-Hall, 1982Google Scholar - [RS]J. H. Reif, J. A. Storer “Shortest Paths in Euclidean Space with Polyhedral Obstacles”, TR CS-85-121, Brandeis University, 1985.Google Scholar
- [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 - [ShS]M. Sharir, A. Schorr “On Shortest Paths in Polyhedral Spaces,”
*SIAM J. Computing, 15,*1, pp. 193–215, 1986.Google Scholar - [Si]E. B. Silverberg, Ph.D. Thesis in preparation, Stanford University, 1986.Google Scholar
- [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

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© Springer-Verlag Berlin Heidelberg 1986