Planar spanners and approximate shortest path queries among obstacles in the plane

  • Srinivasa Arikati
  • Danny Z. Chen
  • L. Paul Chew
  • Gautam Das
  • Michiel Smid
  • Christos D. Zaroliagis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1136)


We consider the problem of finding an obstacle-avoiding path between two points s and t in the plane, amidst a set of disjoint polygonal obstacles with a total of n vertices. The length of this path should be within a small constant factor c of the length of the shortest possible obstacle-avoiding s-t path measured in the L p -metric. Such an approximate shortest path is called a c-short path, or a short path with stretch factor c. The goal is to preprocess the obstacle-scattered plane by creating an efficient data structure that enables fast reporting of a c-short path (or its length). In this paper, we give a family of algorithms for the above problem that achieve an interesting trade-off between the stretch factor, the query time and the preprocessing bounds. Our main results are algorithms that achieve logarithmic length query time, after subquadratic time and space preprocessing.


Short Path Planar Graph Short Path Problem Path Query Short Path Tree 
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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  1. 1.
    S. Arya, G. Das, D.M. Mount, J.S. Salowe and M. Smid, “Euclidean spanners: short, thin, and lanky”, Proc. 27th ACM STOC, 1995, pp. 489–498.Google Scholar
  2. 2.
    S. Arya, D.M. Mount, N.S. Netanyahu, R. Silverman and A. Wu, “An optimal algorithm for approximate nearest neighbor searching”, Proc. 5th ACM-SIAM Symp. on Discrete Algorithms, 1994, pp. 573–582.Google Scholar
  3. 3.
    M. J. Atallah and D. Z. Chen, “Parallel rectilinear shortest paths with rectangular obstacles”, Comp. Geometry: Theory and Appl., 1:2 (1991), pp.79–113.Google Scholar
  4. 4.
    M. J. Atallah and D. Z. Chen, “On parallel rectilinear obstacle-avoiding paths”, Computational Geometry: Theory and Applications, 3 (1993), pp. 307–313.Google Scholar
  5. 5.
    D. Z. Chen. “On the all-pairs Euclidean short path problem”, Proc. 6th Annual ACM-SIAM Symp. on Discrete Algorithms, San Francisco, 1995, pp. 292–301.Google Scholar
  6. 6.
    D. Z. Chen and K. S. Klenk, “Rectilinear short path queries among rectangular obstacles”, Proc. 7th Can. Conf. on Comp. Geometry, 1995, pp. 169–174.Google Scholar
  7. 7.
    D. Z. Chen, K. S. Klenk, and H.-Y. T. Tu, “Rectilinear shortest path queries among weighted obstacles in the rectilinear plane”, Proc. 11th Annual ACM Symp. on Computational Geometry, 1995, pp. 370–379.Google Scholar
  8. 8.
    L. P. Chew, “Constrained Delaunay triangulations”, Algorithmica, 4 (1989), pp. 97–108.CrossRefGoogle Scholar
  9. 9.
    L. P. Chew, “There are planar graphs almost as good as the complete graph”, J. of Computer and System Sciences, 39 (1989), pp. 205–219.Google Scholar
  10. 10.
    L. P. Chew, “Planar graphs and sparse graphs for efficient motion planning in the plane”, Computer Science Tech Report, PCS-TR90-146, Dartmouth College.Google Scholar
  11. 11.
    L. P. Chew and R. L. Drysdale, “Voronoi diagrams based on convex distance functions”, Proc. 1st Annual ACM Symp. on Comp. Geometry, 1985, pp. 235–244.Google Scholar
  12. 12.
    Y.-J. Chiang, F. P. Preparata, and R. Tamassia, “A unified approach to dynamic point location, ray shooting, and shortest paths in planar maps”, Proc. of the 4th ACM-SIAM Symp. on Discrete Algorithms, 1993, pp. 44–53.Google Scholar
  13. 13.
    K. L. Clarkson, “Approximation algorithms for shortest path motion planning”, Proc. 19th Annual ACM Symp. Theory of Computing, 1987, pp. 56–65.Google Scholar
  14. 14.
    H. Djidjev, G. Pantziou and C. Zaroliagis, “On-line and dynamic algorithms for shortest path problems”, Proc. 12th Symp. on Theor. Aspects of Comp. Sc., LNCS 900, Springer-Verlag, 1995, pp. 193–204.Google Scholar
  15. 15.
    D. P. Dobkin, S. J. Friedman, and K. J. Supowit, “Delaunay graphs are almost as good as complete graphs”, Discrete & Comp. Geometry, 5 (1990), pp. 399–407.Google Scholar
  16. 16.
    H. ElGindy and P. Mitra, “Orthogonal shortest route queries among axes parallel rectangular obstacles”, Int. J. of Comp. Geometry and Appl., 4 (1) (1994), 3–24.Google Scholar
  17. 17.
    G. Frederickson, “Fast algorithms for shortest paths in planar graphs, with applications”, SIAM J. on Computing, 16 (1987), pp.1004–1022.Google Scholar
  18. 18.
    G.N. Frederickson, “Using cellular graph embeddings in solving all pairs shortest path problems”, J. of Algorithms, 19 (1995), pp. 45–85.Google Scholar
  19. 19.
    M. Goodrich, “Planar separators and parallel polygon triangulation”, Proc. 24th ACM Symp. on Theory of Comp., 1992, pp.507–516.Google Scholar
  20. 20.
    M.T. Goodrich and R. Tamassia, “Dynamic ray shooting and shortest paths via balanced geodesic triangulations”, Proc. 9th Annual ACM Symp. on Computational Geometry, 1993, pp. 318–327.Google Scholar
  21. 21.
    L.J. Guibas and J. Hershberger, “Optimal shortest path queries in a simple polygon”, Proc. 3rd Annual ACM Symp. on Computational Geometry, 1987, pp. 50–63.Google Scholar
  22. 22.
    L.J. Guibas, J. Hershberger, D. Leven, M. Sharir, and R.E. Tarjan, “Linear time algorithms for visibility and shortest paths problems inside triangulated simple polygons”, Algorithmica, 2 (1987), pp. 209–233.CrossRefGoogle Scholar
  23. 23.
    P. Klein, S. Rao, M. Rauch and S. Subramanian, “Faster shortest-path algorithms for planar graphs”, Proc. 26th ACM Symp. on Theory of Comp., 1994, pp.27–37.Google Scholar
  24. 24.
    R. Lipton and R. Tarjan, “A separator theorem for planar graphs”, SIAM J. Applied Mathematics, 36 (1979), pp.177–189.Google Scholar
  25. 25.
    B. Schieber and U. Vishkin, “On finding lowest common ancestors: Simplification and parallelization”, SIAM J. Computing, 17:6 (1988), pp.1253–1262.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Srinivasa Arikati
    • 1
  • Danny Z. Chen
    • 2
  • L. Paul Chew
    • 3
  • Gautam Das
    • 1
  • Michiel Smid
    • 4
  • Christos D. Zaroliagis
    • 5
  1. 1.Math Sciences DeptThe University of MemphisMemphisUSA
  2. 2.Dept of Computer Sc. and EngUniv. of Notre DameNotre DameUSA
  3. 3.Dept of Computer ScCornell UniversityIthacaUSA
  4. 4.Dept of Computer ScKing's College LondonLondonUK
  5. 5.Max-Planck-Institut für InformatikSaarbrückenGermany

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