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Approximate Euclidean Shortest Paths in Polygonal Domains

Authors R. Inkulu, Sanjiv Kapoor

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Computational Geometry
  • Geometric Shortest Paths
  • Approximation Algorithms

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Abstract

Given a set P of h pairwise disjoint simple polygonal obstacles in R^2 defined with n vertices, we compute a sketch Omega of P whose size is independent of n, depending only on h and the input parameter epsilon. We utilize Omega to compute a (1+epsilon)-approximate geodesic shortest path between the two given points in O(n + h((lg n) + (lg h)^(1+delta) + (1/epsilon) lg(h/epsilon)))) time. Here, epsilon is a user parameter, and delta is a small positive constant (resulting from the time for triangulating the free space of P using the algorithm in [Bar-Yehuda and Chazelle, 1994]). Moreover, we devise a (2+epsilon)-approximation algorithm to answer two-point Euclidean distance queries for the case of convex polygonal obstacles.

Cite As Get BibTex

R. Inkulu and Sanjiv Kapoor. Approximate Euclidean Shortest Paths in Polygonal Domains. In 30th International Symposium on Algorithms and Computation (ISAAC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 149, pp. 11:1-11:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ISAAC.2019.11

Author Details

R. Inkulu
  • Department of Computer Science & Engineering, IIT Guwahati, India
Sanjiv Kapoor
  • Department of Computer Science & Engineering, IIT Chicago, USA

Funding

  • Inkulu, R.: This research is supported in part by SERB MATRICS grant MTR/2017/000474.

References

  1. P. K. Agarwal, R. Sharathkumar, and H. Yu. Approximate Euclidean shortest paths amid convex obstacles. In Proceedings of Symposium on Discrete Algorithms, pages 283-292, 2009. Google Scholar
  2. S. R. Arikati, D. Z. Chen, L. P. Chew, G. Das, M. H. M. Smid, and C. D. Zaroliagis. Planar spanners and approximate shortest path queries among obstacles in the plane. In Proceedings of European Symposium on Algorithms, pages 514-528, 1996. Google Scholar
  3. R. Bar-Yehuda and B. Chazelle. Triangulating disjoint Jordan chains. International Journal of Computational Geometry & Applications, 4(4):475-481, 1994. Google Scholar
  4. D. Z. Chen. On the all-pairs Euclidean short path problem. In Proceedings of Symposium on Discrete Algorithms, pages 292-301, 1995. Google Scholar
  5. D. Z. Chen and H. Wang. Computing shortest paths among curved obstacles in the plane. ACM Transactions on Algorithms, 11(4):26:1-26:46, 2015. Google Scholar
  6. L. P. Chew. There are planar graphs almost as good as the complete graph. Journal of Computer and System Sciences, 39(2):205-219, 1989. Google Scholar
  7. Y.-J. Chiang and J. S. B. Mitchell. Two-point Euclidean shortest path queries in the plane. In Proceedings of Symposium on Discrete Algorithms, pages 215-224, 1999. Google Scholar
  8. K. L. Clarkson. Approximation algorithms for shortest path motion planning. In Proceedings of Symposium on Theory of Computing, pages 56-65, 1987. Google Scholar
  9. T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein. Introduction to Algorithms. The MIT Press, 2009. Google Scholar
  10. Toth C. D., O'Rourke J., and Goodman J. E. Handbook of discrete and computational geometry. CRC Press, 3rd ed. edition, 2017. Google Scholar
  11. M. de Berg, O. Cheong, M. van Kreveld, and M. Overmars. Computational Geometry: algorithms and applications. Springer-Verlag, 3rd ed. edition, 2008. Google Scholar
  12. D. P. Dobkin and David G. Kirkpatrick. Fast detection of polyhedral intersection. Theoretical Computer Science, 27:241-253, 1983. Google Scholar
  13. M. L. Fredman and R. E. Tarjan. Fibonacci heaps and their uses in improved network optimization algorithms. Journal of ACM, 34(3):596-615, 1987. Google Scholar
  14. S. K. Ghosh. Visibility algorithms in the plane. Cambridge University Press, New York, USA, 2007. Google Scholar
  15. L. J. Guibas and J. Hershberger. Optimal shortest path queries in a simple polygon. Journal of Computer and System Sciences, 39(2):126-152, 1989. Google Scholar
  16. H. Guo, A. Maheshwari, and J-R. Sack. Shortest path queries in polygonal domains. In Proceedings of Algorithmic Aspects in Information and Management, pages 200-211, 2008. Google Scholar
  17. J. Hershberger and S. Suri. An optimal algorithm for Euclidean shortest paths in the plane. SIAM Journal on Computing, 28(6):2215-2256, 1999. Google Scholar
  18. R. Inkulu, S. Kapoor, and S. N. Maheshwari. A near optimal algorithm for finding Euclidean shortest path in polygonal domain. CoRR, abs/1011.6481, 2010. URL: http://arxiv.org/abs/1011.6481.
  19. S. Kapoor. Efficient computation of geodesic shortest paths. In Proceedings of Symposium on Theory of Computing, pages 770-779, 1999. Google Scholar
  20. S. Kapoor and S. N. Maheshwari. Efficent algorithms for Euclidean shortest path and visibility problems with polygonal obstacles. In Proceedings of Symposium on Computational Geometry, pages 172-182, 1988. Google Scholar
  21. S. Kapoor and S. N. Maheshwari. Efficiently constructing the visibility graph of a simple polygon with obstacles. SIAM Jounral on Computing, 30(3):847-871, 2000. Google Scholar
  22. S. Kapoor, S. N. Maheshwari, and J. S. B. Mitchell. An efficient algorithm for Euclidean shortest paths among polygonal obstacles in the plane. Discrete & Computational Geometry, 18(4):377-383, 1997. Google Scholar
  23. K. Kawarabayashi, P. N. Klein, and C. Sommer. Linear-space approximate distance oracles for planar, bounded-genus and minor-free graphs. In Proceedings of Colloquium on Automata, Languages and Programming, pages 135-146, 2011. Google Scholar
  24. D. G. Kirkpatrick. Optimal search in planar subdivisions. SIAM Journal on Computing, 12(1):28-35, 1983. Google Scholar
  25. J. Kleinberg and E. Tardos. Algorithm Design. Addison-Wesley Longman Publishing Co., Inc., 2005. Google Scholar
  26. J. S. B. Mitchell. Shortest paths among obstacles in the plane. International Journal of Computational Geometry & Applications, 6(3):309-332, 1996. Google Scholar
  27. J. S. B. Mitchell. Geometric shortest paths and network optimization. In J.-R. Sack and J. Urrutia, editors, Handbook of Computational Geometry, pages 633-701. North-Holland, 2000. Google Scholar
  28. S. Sen. Approximating shortest paths in graphs. In Proceedings of Workshop on Algorithms and Computation, pages 32-43, 2009. Google Scholar
  29. E. Welzl. Constructing the visibility graph for n-line segments in O(n²) time. Information Processing Letters, 20(4):167-171, 1985. Google Scholar
  30. A. C. Yao. On constructing minimum spanning trees in k-dimensional spaces and related problems. SIAM Journal on Computing, 11(4):721-736, 1982. Google Scholar
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