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3-dimensional shortest paths in the presence of polyhedral obstacles

  • John H. Reif
  • James A. Storer
Invited Lectures
Part of the Lecture Notes in Computer Science book series (LNCS, volume 324)

Abstract

We consider the problem of finding a minimum length path between two points in 3-dimensional Euclidean space which avoids a set of (not necessarily convex) polyhedral obstacles; we let n denote the number of the obstacle edges and k denote the number of "islands" in the obstacle space. An island is defined to be a maximal convex obstacle surface such that for any two points contained in the interior of the island, a minimal length path between these two points is strictly contained in the interior of the island; for example, a set of i disconnected convex polyhedra forms a set of i islands, however, a single non-convex polyhedron will constitute more that one island. Prior to this work, the best known algorithm required double-exponential time. We present an algorithm that runs in \(n^{k^{0(1)} }\) time and also one that runs in O(nlog(k)) space.

Keywords

Mover's problem minimal movement problem shortest path Euclidean space robotics motion planning Voronoi diagram quadratic curves theory of real closed fields 

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References

  1. C. Bajaj [1984]. "Reducibility among Geometric Location-Allocation Optimization Problems", Technical Report TR84-607, Computer Science Dept., Cornell University, Ithaca, NY.Google Scholar
  2. M. Ben-Or, D. Kozen, and J. Reif [1984]. "The Complexity of Elementary Algebra and Geometry", Proceedings 16 th ACM Symposium on the Theory of Computing, Washington, DC, 457–464.Google Scholar
  3. J. Canny and J. H. Reif [1987]. "New Lower Bounds for Robot Motion Planning Problems", Proceedings 28th Annual IEEE Symposium on the Foundations of Computer Science, Los Angeles, CA, 49–60.Google Scholar
  4. A. L. Chistov and D. Yu. Grigor'ev [1985]. "Complexity of Quantifier Elimination in the Theory of Algebraically Closed Fields", Technical Report, Institute of the Academy of Sciences, Leningrad, USSR.Google Scholar
  5. G. E. Collins [1975]. "Quantifier Elimination for Real Closed Fields by Cylindric Algebraic Decomposition", Proceedings Second GI Conference on Automata Theory and Formal Languages, Springer-Verlag LNCS 35, Berlin, 134–183.Google Scholar
  6. W. R. Franklin and V. Akman [1984]. "Minimal Paths Between Two Points On/Around A Convex Polyhedron", Technical Report, Electrical, Computer, and Systems Engineering Dept., Rensselaer Polytechnic Institute, Troy, NY.Google Scholar
  7. W. R. Franklin, V. Akman, and C. Verrilli [1984]. "Voronoi Diagrams with Barriers and on Polyhedra", Technical Report, Electrical, Computer, and Systems Engineering Dept., Rensselaer Polytechnic Institute, Troy, NY.Google Scholar
  8. J. E. Hopcroft, D. A. Joseph, and S. H. Whitesides [1982]. "On the Movement of Robot Arms in 2-Dimensional Bounded Regions", Proceedings 23rd IEEE Symposium on Foundations of Computer Science", Chicago, IL, 280–289.Google Scholar
  9. J. E. Hopcroft, D. A. Joseph, and S. H. Whitesides [1982b]. "Movement Problems for 2-Dimensional Linkages", Technical Report TR82-515, Computer Science Dept., Cornell University, Ithaca, NY.Google Scholar
  10. J. E. Hopcroft, D. A. Joseph, and S. H. Whitesides [1982c]. "Determining Points of a Circular Region Reachable by Joints of a Robot Arm", Technical Report TR82-516, Computer Science Dept., Cornell University, Ithaca, NY.Google Scholar
  11. J. E. Hopcroft and G. Wilfong [1984]. "On the Motion of Objects in Contact", Technical Report TR84-602, Computer Science Dept., Cornell University, Ithaca, NY.Google Scholar
  12. J. E. Hopcroft and G. Wilfong [1984b]. "Reducing Multiple Object Motion Planning to Graph Searching", Technical Report TR84-616, Computer Science Dept., Cornell University, Ithaca, NY.Google Scholar
  13. D. A. Joseph and W. H. Plantinga [1985]. "On the Complexity of Reachability and Motion Planning Questions", Proceedings First Annual Conference of Computation Geometry, Baltimore, MD, 62–66.Google Scholar
  14. K. Kedem and M. Sharir [1985]. "An Efficient Algorithm for Planning Collision-Free Translational Motion of a Convex Polyhedral Object in 2-Dimensional Space Admidst Polygonal Obstacles", Proceedings First Annual Conference of Computational Geometry, Baltimore, MD, 75–80.Google Scholar
  15. D. Leven and M. Sharir [1985]. "An Efficient and Simple Motion Planning Algorithm for a Ladder Moving in Two-Dimensional Space Admidst Polygonal Barriers", Proceedings First Annual Conference of Computational Geometry, Baltimore, MD, 221–227.Google Scholar
  16. T. Lozano-Perez [1980]. "Automatic Planning of Manipulation Transfer Movements", AI Memo 606, Artificial Intelligence Laboratory, MIT, Cambridge, MA.Google Scholar
  17. T. Lozano-Perez and M. A. Wesley [1979]. "An Algorithm for Planning Collision-Free Paths among Polyhedral Obstacles", CACM 22:10, 560–570.Google Scholar
  18. C. O'Dunlaing, M. Sharir, and C. K. Yap [1983]. "Retraction: A New Approach to Motion Planning", Proceedings 15 th ACM Symposium on the Theory of Computing, Boston, MA, 207–220.Google Scholar
  19. J. O'Rourke, S. Suri, and H. Booth [1984]. "Shortest Paths on Polyhedral Surfaces", Technical Report, Dept. of Electrical Engineering and Computer Science, Johns Hopkins University.Google Scholar
  20. C. H. Papadimitriou [1984]. "An Algorithm for Shortest Path Motion in Three Dimensions", Technical Report, Stanford University.Google Scholar
  21. J. Reif [1979]. "Complexity of the Mover's Problem", Proceedings 20 th IEEE Symposium on Foundations of Computer Science", San Juan, Puerto Rico, 421–427; also to appear in Planning, Geometry, and the Complexity of Robot Planning, ed. by J. Schwartz.Google Scholar
  22. J. T. Schwartz and M. Sharir [1981]. "On the Piano Movers Problem 1: The Case of a 2-Dimensional Rigid Polygonal Body Moving Amidst Barriers", Technical Report TR39, Computer Science Dept., New York University, NY.Google Scholar
  23. J. T. Schwartz and M. Sharir [1982]. "On the Piano Movers Problem 2: General Techniques for Computing Topological Properties of Real Algebraic Manifolds", Technical Report TR41, Computer Science Dept., New York University, NY.Google Scholar
  24. M. Sharir and A. Schorr [1984]. "On Shortest Paths in Polyhedral Spaces", Proceedings 16 th ACM Symposium on the Theory of Computing, Washington, DC, 144–153.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • John H. Reif
    • 1
  • James A. Storer
    • 2
  1. 1.Computer Science DepartmentDuke UniversityDurham
  2. 2.Computer Science DepartmentBrandeis UniversityWaltham

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