Computing Shortest Paths in the Plane with Removable Obstacles

Authors Pankaj K. Agarwal, Neeraj Kumar, Stavros Sintos, Subhash Suri



PDF
Thumbnail PDF

File

LIPIcs.SWAT.2018.5.pdf
  • Filesize: 0.63 MB
  • 15 pages

Document Identifiers

Author Details

Pankaj K. Agarwal
  • Duke University, Durham, NC, USA
Neeraj Kumar
  • University of California, Santa Barbara, CA, USA
Stavros Sintos
  • Duke University, Durham, NC, USA
Subhash Suri
  • University of California, Santa Barbara, CA, USA

Cite AsGet BibTex

Pankaj K. Agarwal, Neeraj Kumar, Stavros Sintos, and Subhash Suri. Computing Shortest Paths in the Plane with Removable Obstacles. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 5:1-5:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.SWAT.2018.5

Abstract

We consider the problem of computing a Euclidean shortest path in the presence of removable obstacles in the plane. In particular, we have a collection of pairwise-disjoint polygonal obstacles, each of which may be removed at some cost c_i > 0. Given a cost budget C > 0, and a pair of points s, t, which obstacles should be removed to minimize the path length from s to t in the remaining workspace? We show that this problem is NP-hard even if the obstacles are vertical line segments. Our main result is a fully-polynomial time approximation scheme (FPTAS) for the case of convex polygons. Specifically, we compute an (1 + epsilon)-approximate shortest path in time O({nh}/{epsilon^2} log n log n/epsilon) with removal cost at most (1+epsilon)C, where h is the number of obstacles, n is the total number of obstacle vertices, and epsilon in (0, 1) is a user-specified parameter. Our approximation scheme also solves a shortest path problem for a stochastic model of obstacles, where each obstacle's presence is an independent event with a known probability. Finally, we also present a data structure that can answer s-t path queries in polylogarithmic time, for any pair of points s, t in the plane.

Subject Classification

ACM Subject Classification
  • Theory of computation → Shortest paths
Keywords
  • Euclidean shortest paths
  • Removable polygonal obstacles
  • Stochastic shortest paths
  • L_1 shortest paths

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Esther M Arkin, Joseph SB Mitchell, and Christine D Piatko. Bicriteria shortest path problems in the plane. In Proc. 3rd Canad. Conf. Comput. Geom, pages 153-156, 1991. Google Scholar
  2. T. M. Chan. Low-dimensional linear programming with violations. SIAM J. Comput., 34(4):879-893, 2005. Google Scholar
  3. T. M. Chan and Y. Nekrich. Towards an optimal method for dynamic planar point location. In Proc. 56th Symp. Found. Comp. Science, pages 390-409. IEEE, 2015. Google Scholar
  4. D. Z. Chen, K. S. Klenk, and H. T. Tu. Shortest path queries among weighted obstacles in the rectilinear plane. SIAM J. Comput., 29(4):1223-1246, 2000. Google Scholar
  5. Danny Z Chen, Ovidiu Daescu, and Kevin S Klenk. On geometric path query problems. Int. J. Comp. Geom. &Applic., 11(06):617-645, 2001. Google Scholar
  6. Danny Z Chen, John Hershberger, and Haitao Wang. Computing shortest paths amid convex pseudodisks. SIAM J. Comput., 42(3):1158-1184, 2013. Google Scholar
  7. Danny Z Chen, Rajasekhar Inkulu, and Haitao Wang. Two-point L₁ shortest path queries in the plane. In Proc. 30th Annual Symp. Comput. Geom., page 406. ACM, 2014. Google Scholar
  8. Danny Z Chen and Haitao Wang. A nearly optimal algorithm for finding L₁ shortest paths among polygonal obstacles in the plane. In Proc. 19th Europ. Symp. Alg., pages 481-492. Springer, 2011. Google Scholar
  9. Danny Z Chen and Haitao Wang. L₁ shortest path queries among polygonal obstacles in the plane. In Proc. 30th Int. Symp. Theor. Asp. Comp. Science, volume 20, 2013. Google Scholar
  10. Danny Z Chen and Haitao Wang. Computing shortest paths among curved obstacles in the plane. ACM Transactions on Algorithms, 11(4):26, 2015. Google Scholar
  11. Danny Z Chen and Haitao Wang. A new algorithm for computing visibility graphs of polygonal obstacles in the plane. J. Comput. Geom., 6(1):316-345, 2015. Google Scholar
  12. Y.J Chiang and J.S.B Mitchell. Two-point Euclidean shortest path queries in the plane. In Proc. 10th ACM-SIAM Annual Symp. Discrete Algorithms. SIAM, 1999. Google Scholar
  13. K. Clarkson, S. Kapoor, and P. Vaidya. Rectilinear shortest paths through polygonal obstacles in O(n log² n) time. In Proc. 3rd Annual Symp. Comput. Geom., pages 251-257. ACM, 1987. Google Scholar
  14. T. Feder, R. Motwani, L. O'Callaghan, C. Olston, and R. Panigrahy. Computing shortest paths with uncertainty. J. Algorithms, 62(1):1-18, 2007. Google Scholar
  15. Y. Gao. Shortest path problem with uncertain arc lengths. Computers &Mathematics with Applications, 62(6):2591-2600, 2011. Google Scholar
  16. Subir Kumar Ghosh. Visibility algorithms in the plane. Cambridge university press, 2007. Google Scholar
  17. S. Har-Peled and V. Koltun. Separability with outliers. In Proc. Int. Symp. Alg. and Comput., pages 28-39. Springer, 2005. Google Scholar
  18. J. Hershberger, N. Kumar, and S. Suri. Shortest paths in the plane with obstacle violations. In Proc. 25th Annual Eur. Symp. on Alg., volume 87, pages 49:1-49:14, 2017. Google Scholar
  19. J. Hershberger and S. Suri. An optimal algorithm for Euclidean shortest paths in the plane. SIAM J. Comput., 28(6):2215-2256, 1999. Google Scholar
  20. Rajasekhar Inkulu and Sanjiv Kapoor. Planar rectilinear shortest path computation using corridors. J. Comput. Geom., 42(9):873-884, 2009. Google Scholar
  21. M Iwai, H Suzuki, and T Nishizeki. Shortest path algorithm in the plane with rectilinear polygonal obstacles. In Proc. SIGAL Workshop, 1994. Google Scholar
  22. P. Kamousi, T M Chan, and S. Suri. Stochastic minimum spanning trees in Euclidean spaces. In Proc. 27th Annual Symp. Comput. Geom., pages 65-74. ACM, 2011. Google Scholar
  23. D.T Lee, C.-D. Yang, and T.H Chen. Shortest rectilinear paths among weighted obstacle. Int. J. Comput. Geom. &Appl., 1(02):109-124, 1991. Google Scholar
  24. D.T Lee, C.D Yang, and C.K. Wong. Rectilinear paths among rectilinear obstacles. Discrete Applied Mathematics, 70(3):185-215, 1996. Google Scholar
  25. A. Maheshwari, S. C. Nandy, D. Pattanayak, S. Roy, and M. Smid. Geometric path problems with violations. Algorithmica, pages 1-24, 2016. Google Scholar
  26. J. Matoušek. On geometric optimization with few violated constraints. Discrete &Computational Geometry, 14(4):365-384, 1995. Google Scholar
  27. J. S. B. Mitchell and C. H. Papadimitriou. The weighted region problem: finding shortest paths through a weighted planar subdivision. J. ACM, 38(1):18-73, 1991. Google Scholar
  28. Joseph S.B. Mitchell. Geometric shortest paths and network optimization. In Handbook of Computational Geometry, pages 633-701. Elsevier Science Publishers B.V. North-Holland, 1998. Google Scholar
  29. Evdokia Nikolova, Matthew Brand, and David R Karger. Optimal route planning under uncertainty. In Proc. 16th Int. Conf. Autom. Plann. and Sched., volume 6, pages 131-141, 2006. Google Scholar
  30. T. Roos and P. Widmayer. k-violation linear programming. Inf. Process. Lett., 52(2):109-114, 1994. Google Scholar
  31. Jörg-Rüdiger Sack and Jorge Urrutia. Handbook of computational geometry. Elsevier, 1999. Google Scholar
  32. Neil Sarnak and Robert E Tarjan. Planar point location using persistent search trees. Communic. ACM, 29(7):669-679, 1986. Google Scholar
  33. H. Wang. Bicriteria rectilinear shortest paths among rectilinear obstacles in the plane. In Proc. 33rd Annual Symp. Comput. Geom., pages 60:1-60:16, 2017. Google Scholar
  34. C.D Yang, D.T. Lee, and C.K Wong. On bends and lengths of rectilinear paths: a graph-theoretic approach. Int. J. Comput. Geom. &Appl., 2(01):61-74, 1992. Google Scholar
  35. C.D Yang, D.T. Lee, and C.K. Wong. Rectilinear path problems among rectilinear obstacles revisited. SIAM J. Comput., 24(3):457-472, 1995. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail