Skip to main content
SpringerLink
Log in

Visibility graphs and obstacle-avoiding shortest paths

  • Published:
Zeitschrift für Operations-Research Aims and scope Submit manuscript

Summary

Two closely related problems in Computational Geometry are determining visibility graphs and shortest paths in a two- or three-dimensional environment containing obstacles. Applications are within Computer Graphics and Robotics. We give a survey on recent research done on efficient algorithms for these problems.

Zusammenfassung

Zwei eng miteinander verwandte Probleme in der algorithmischen Geometrie sind die Bestimmung von Sichtbarkeitsgraphen und kürzesten Wegen in einer zwei- oder dreidimensionalen Umgebung mit Hindernissen. Anwendungen finden sich insbesondere auf den Gebieten Computergrafik und Robotik. Diese Arbeit gibt einen Überblick über kürzlich erschienene Arbeiten zum Entwurf effizienter Algorithmen für diese Probleme.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aho AV, Hopcroft JE, Ullman JD (1974) The design and analysis of computer algorithms. Addison-Wesley

  2. Asano T, Asano T, Guibas L, Hershberger J, Imai H (1986) Visibility of disjoint polygons. Algorithmica 1:49–63

    Article  MathSciNet  MATH  Google Scholar 

  3. Baker B (1985) Shortest paths with unit clearance among polygonal obstacles. SIAM Conference on Geometrical Modelling and Robotics

  4. Canny J (1987) A new algebraic method for robot motion planning and real geometry. In: Proceedings of the 28th IEEE Symposium on Foundations of Computer Science, pp 39–48

  5. Canny J, Reif J (1987) New lower bound techniques for robot motion planning problems. In: Proceedings of the 28th IEEE Symposium on Foundations of Computer Science, pp 49–60

  6. Chew LP (1987) Planar graphs and sparse graphs for efficient motion planning in the plane. Manuscript

  7. Chew LP (1985) Planning the shortest path for a disk inO(n 2 logn) time. In: Proceedings of the 1st Annual Symposium on Computational Geometry, pp 214–220

  8. Clarkson K (1987) Approximation algorithms for shortest path motion planning. In: Proceedings of the 19th Annual ACM Symposium on Theory of Computing, pp 56–65

  9. Clarkson K, Kapoor S, Vaidya P (1987) Rectilinear shortest paths through polygonal obstacles inO(n log2 n) time. In: Proceedings of the 3rd Annual Symposium on Computational Geometry, pp 251–257

  10. Collins GE (1975) Quantifier elimination for real closed fields by cylindric algebraic decomposition. In: Proceedings 2nd GI-Conference on Automata Theory and Formal Languages, Lecture Notes in Computer Science 35, Springer-Verlag, pp 134–183

  11. Edelsbrunner H (1987) Algorithms in combinatorial geometry. Springer-Verlag

  12. Edelsbrunner H, Guibas L (1986) Topologically sweeping in an arrangement. In: Proceedings of the 18th ACM Symposium on Theory of Consumpting, pp 389–403

  13. Fredman M, Tarjan R (1987) Fibonacci heaps and their uses in improved network optimization algorithms. J of the Association for Computing Machinery 34:596–615

    Article  MathSciNet  Google Scholar 

  14. Ghosh SK, Mount DM (1987) An output sensitive algorithm for computing visibility graphs. In: Proceedings of the 28th Annual Symposium on Foundations of Computer Science, pp 11–19

  15. Guibas LJ, Hershberger J (1987) Optimal shortest path queries in a simple polygon. In: Proceedings of the 9rd Annual Symposium on Computational Geometry, pp 50–63

  16. Hershberger JE (1987) Efficient algorithms for shortest path and visibility problems. Dissertation, Stanford University

  17. Hershberger JE, Guibas LJ (1986) AnO(n 2) shortest path algorithm for a non-rotating convex body. Technical Report, DEC Systems Research Center

  18. Lee DT (1978) Proximity and reachability in the plane. Dissertation, University of Illinois at Urbana-Champaign

    Google Scholar 

  19. Lee DT, Preparata FP (1984) Euclidean shortest paths in the presence of rectilinear barriers. Networks 14:393–410

    Article  MathSciNet  MATH  Google Scholar 

  20. Lozano-Perez T, Wesley MA (1979) An algorithm for planning collision-free paths among polyhedral obstacles. Communications of the ACM 22:560–570

    Article  Google Scholar 

  21. Masser M (1988) Algorithmen zur Konstruktion von Sichtbarkeitsgraphen. Diplomarbeit, Institutes for Information Processing, IIG, Technical University of Graz, in preparation

  22. Mehlhorn K (1984) Data structures and algorithms 2: graph algorithms and NP-completeness. Springer-Verlag

  23. Mehlhorn K (1984) Data structures and algorithms 3: multidimensional searching and computational geometry. Springer-Verlag

  24. Mitchell JSB, Papadimitriou CH (1987) The weighted region problem. In: Proceedings of the 3rd Annual Symposium on Computational Geometry, pp 30–38

  25. Mitchel JSB, Mount DM, Papadimitriou CH (1987) The discrete geodesic problem. SIAM J Computing 23:647–668

    Article  MathSciNet  MATH  Google Scholar 

  26. Overmars M, Welzl E (1987) Construction of sparse visibility graphs. Report RUU-CS-87-9, Department of Computer Science, University of Utrecht

  27. Overmars M, Welzl E (1988) New methods for computing visibility graphs. To appear in: Proceedings of the 4th Annual Symposium on Computational Geometry

  28. Papadimitriou CH (1985) An algorithm for shortest path motion in three dimensions. Inform Process Lett 20:259–263

    Article  MathSciNet  MATH  Google Scholar 

  29. Papadimitriou CH, Silverberg EB (1986) Optimal piecewise linear motion of an object among obstacles. Technical Report, Dept. of Operations Research, Stanford University

  30. Preparata FP, Shamos MI (1985) Computational geometry. Springer Verlag

  31. Reif J, Storer JA (1985) Shortest paths in Euclidean space with polyhedral obstacles. Technical Report CS-85-121, Computer Science Department, Brandeis University

  32. Rohner H (1986) Shortest paths in the plane with convex polygonal obstacles. Inform Process Lett 23:71–76

    Article  MathSciNet  Google Scholar 

  33. Schorr A, Sharir M (1986) On shortest paths in polyhedral spaces. SIAM J Computing 15:193–215

    Article  MathSciNet  MATH  Google Scholar 

  34. Tarjan RE, van Wyk C (1986) AnO(n log log n)-time algorithm for triangulating simple polygons. Manuscript

  35. Welzl E (1985) Constructing the visibility graph forn line segments inO(n 2) time. Inform Process Lett 20:167–171

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich Mathematik, Freie Universität Berlin, Arnimallee 2-6, D-1000, Berlin 33, FRG

    H. Alt & E. Welzl

Authors
  1. H. Alt
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Welzl
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Alt, H., Welzl, E. Visibility graphs and obstacle-avoiding shortest paths. Zeitschrift für Operations Research 32, 145–164 (1988). https://doi.org/10.1007/BF01928918

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01928918

Key words

Search

Navigation