A Divide-and-Conquer Algorithm for Two-Point L_1 Shortest Path Queries in Polygonal Domains

Author Haitao Wang



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Haitao Wang
  • Department of Computer Science, Utah State University, Logan, UT 84322, USA

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Haitao Wang. A Divide-and-Conquer Algorithm for Two-Point L_1 Shortest Path Queries in Polygonal Domains. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 59:1-59:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.SoCG.2019.59

Abstract

Let P be a polygonal domain of h holes and n vertices. We study the problem of constructing a data structure that can compute a shortest path between s and t in P under the L_1 metric for any two query points s and t. To do so, a standard approach is to first find a set of n_s "gateways" for s and a set of n_t "gateways" for t such that there exist a shortest s-t path containing a gateway of s and a gateway of t, and then compute a shortest s-t path using these gateways. Previous algorithms all take quadratic O(n_s * n_t) time to solve this problem. In this paper, we propose a divide-and-conquer technique that solves the problem in O(n_s + n_t log n_s) time. As a consequence, we construct a data structure of O(n+(h^2 log^3 h/log log h)) size in O(n+(h^2 log^4 h/log log h)) time such that each query can be answered in O(log n) time.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Computational geometry
Keywords
  • shortest paths
  • two-point queries
  • L_1 metric
  • polygonal domains

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References

  1. A. Aggarwal, M. Klawe, S. Moran, P. Shor, and R. Wilbur. Geometric Applications of a Matrix-Searching Algorithm. Algorithmica, 2:195-208, 1987. Google Scholar
  2. S.W. Bae and H. Wang. L₁ shortest path queries in simple polygons, 2018. URL: http://arxiv.org/abs/1809.07481.
  3. R. Bar-Yehuda and B. Chazelle. Triangulating disjoint Jordan chains. International Journal of Computational Geometry and Applications, 4(4):475-481, 1994. Google Scholar
  4. B. Chazelle. Triangulating a Simple Polygon in Linear Time. Discrete and Computational Geometry, 6:485-524, 1991. Google Scholar
  5. D.Z. Chen, O. Daescu, and K.S. Klenk. On geometric path query problems. International Journal of Computational Geometry and Applications, 11(6):617-645, 2001. Google Scholar
  6. D.Z. Chen, R. Inkulu, and H. Wang. Two-Point L₁ Shortest Path Queries in the Plane. Journal of Computational Geometry, 1:473-519, 2016. Google Scholar
  7. D.Z. Chen, K.S. Klenk, and H.-Y.T. Tu. Shortest path queries among weighted obstacles in the rectilinear plane. SIAM Journal on Computing, 29(4):1223-1246, 2000. Google Scholar
  8. D.Z. Chen and H. Wang. Computing L₁ Shortest Paths Among Polygonal Obstacles in the Plane. Algorithmica, 2019. https://doi.org/10.1007/s00453-018-00540-x. Google Scholar
  9. D.Z. Chen and J. Xu. Shortest path queries in planar graphs. In Proceedings of the 32nd Annual ACM Symposium on Theory of Computing (STOC), pages 469-478, 2000. Google Scholar
  10. Y.-J. Chiang and J.S.B. Mitchell. Two-point Euclidean shortest path queries in the plane. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 215-224, 1999. Google Scholar
  11. K. Clarkson, S. Kapoor, and P. Vaidya. Rectilinear shortest paths through polygonal obstacles in O(n log² n) time. In Proceedings of the 3rd Annual Symposium on Computational Geometry (SoCG), pages 251-257, 1987. Google Scholar
  12. K. Clarkson, S. Kapoor, and P. Vaidya. Rectilinear shortest paths through polygonal obstacles in O(n log^2/3 n) time. Manuscript, 1988. Google Scholar
  13. H.N. Djidjev. Efficient algorithms for shortest path queries in planar digraphs. In Proceedings of the 22nd International Workshop on Graph-Theoretic Concepts in Computer Science, pages 151-165, 1996. Google Scholar
  14. H.A. ElGindy and P. Mitra. Orthogonal shortest route queries among axis parallel rectangular obstacles. International Journal of Computational Geometry and Application, 4:3-24, 1994. Google Scholar
  15. J. Fakcharoenphol and S. Rao. Planar graphs, negative weight edges, shortest paths, and near linear time. Journal of Computer and System Sciences, 72:868-889, 2006. Google Scholar
  16. P. Gawrychowski, S. Mozes, O. Weimann, and C. Wulff-Nilsen. Better tradeoffs for exact distance oracles in planar graphs. In Proceedings of the 29rd Annual Symposium on Discrete Algorithms (SODA), pages 515-529, 2018. Google Scholar
  17. 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
  18. H. Guo, A. Maheshwari, and J.-R. Sack. Shortest Path Queries in Polygonal Domains. In Proceedings of the 4th International Conference on Algorithmic Aspects in Information and Management (AAIM), pages 200-211, 2008. Google Scholar
  19. J. Hershberger. A new data structure for shortest path queries in a simple polygon. Information Processing Letters, 38(5):231-235, 1991. Google Scholar
  20. J. Hershberger and J. Snoeyink. Computing minimum length paths of a given homotopy class. Computational Geometry: Theory and Applications, 4(2):63-97, 1994. Google Scholar
  21. 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
  22. R. Inkulu and S. Kapoor. Planar rectilinear shortest path computation using corridors. Computational Geometry: Theory and Applications, 42(9):873-884, 2009. Google Scholar
  23. M.M. Klawe and D.J. Kleitman. An Almost Linear Time Algorithm for Generalized Matrix Searching. SIAM Journal on Discrete Mathematics, 3:81-97, 1990. Google Scholar
  24. J.S.B. Mitchell. An optimal algorithm for shortest rectilinear paths among obstacles. Abstracts of the 1st Canadian Conference on Computational Geometry, 1989. Google Scholar
  25. J.S.B. Mitchell. L₁ shortest paths among polygonal obstacles in the plane. Algorithmica, 8(1):55-88, 1992. Google Scholar
  26. S. Mozes and C. Sommer. Exact distance oracles for planar graphs. In Proceedings of the 23rd Annual Symposium on Discrete Algorithms (SODA), pages 209-222, 2012. Google Scholar
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