Advertisement

Navigating Weighted Regions with Scattered Skinny Tetrahedra

  • Siu-Wing Cheng
  • Man-Kwun Chiu
  • Jiongxin Jin
  • Antoine Vigneron
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9472)

Abstract

We propose an algorithm for finding a \((1+\varepsilon )\)-approximate shortest path through a weighted 3D simplicial complex \(\mathcal T\). The weights are integers from the range [1, W] and the vertices have integral coordinates. Let N be the largest vertex coordinate magnitude, and let n be the number of tetrahedra in \(\mathcal T\). Let \(\rho \) be some arbitrary constant. Let \(\kappa \) be the size of the largest connected component of tetrahedra whose aspect ratios exceed \(\rho \). There exists a constant C dependent on \(\rho \) but independent of \(\mathcal T\) such that if \(\kappa \le \frac{1}{C}\log \log n + O(1)\), the running time of our algorithm is polynomial in n, \(1/\varepsilon \) and \(\log (NW)\). If \(\kappa = O(1)\), the running time reduces to \(O(n \varepsilon ^{-O(1)}(\log (NW))^{O(1)})\).

Keywords

Weighted region Shortest path Approximation algorithm 

References

  1. 1.
    Ahmed, M.: Constrained Shortest Paths in Terrains and Graphs. Ph.D. Thesis, University of Waterloo, Canada (2009)Google Scholar
  2. 2.
    Aleksandrov, L., Maheshwari, A., Sack, J.-R.: Determining approximate shortest paths on weighted polyhedral surfaces. J. ACM 52, 25–53 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Aleksandrov, L., Djidjev, H., Maheshwari, A., Sack, J.-R.: An approximation algorithm for computing shortest paths in weighted 3-d domains. Discrete. Comput. Geom. 50, 124–184 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, J., Han, Y.: Shortest paths on a polyhedron. Int. J. Comput. Geom. Appl. 6, 127–144 (1996)CrossRefzbMATHGoogle Scholar
  5. 5.
    Cheng, S.-W., Jin, J.: Approximate shortest descending paths. SIAM J. Comput. 43, 410–428 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cheng, S.-W., Jin, J.: Shortest paths on polyhedral surfaces and terrains. In: Proceedings of ACM Sympoisum on Theory of Computing, pp. 373–382 (2014)Google Scholar
  7. 7.
    Cheng, S.-W., Jin, J., Vigneron, A.: Triangulation refinement and approximate shortest paths in weighted regions. In: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms, pp. 1626–1640 (2015)Google Scholar
  8. 8.
    Cheng, S.-W., Na, H.-S., Vigneron, A., Wang, Y.: Approximate shortest paths in anisotropic regions. SIAM J. Comput. 38, 802–824 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cheng, S.-W., Na, H.-S., Vigneron, A., Wang, Y.: Querying approximate shortest paths in anisotropic regions. SIAM J. Comput. 39, 1888–1918 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Choi, J., Sellen, J., Yap, C.-K.: Approximate Euclidean shortest path in 3-space. In: Proceedings of the Annual Symposium on Computational Geometry, pp. 41–48 (1994)Google Scholar
  11. 11.
    Clarkson, K.L.: Approximation algorithms for shortest path motion planning. In: Proceedings of the ACM Symposium on Theory Computing, pp. 56–65 (1987)Google Scholar
  12. 12.
    Hershberger, J., Subhash, S.: An optimal algorithm for Euclidean shortest paths in the plane. SIAM J. Comput. 28, 2215–2256 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Menke, W.: Geophysical Data Analysis: Discrete Inverse Theory. Academic Press, New York (2012)zbMATHGoogle Scholar
  14. 14.
    Mitchell, J.S.B., Papadimitrou, C.H.: The weighted region problem: finding shortest paths through a weighted planar subdivision. J. ACM 8, 18–73 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Papadimitriou, C.H.: An algorithm for shortest-path motion in three dimensions. Inf. Process. Lett. 20, 259–263 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Papadimitriou, C.H., Steiglitz, K.: Combinatorial Optimization: Algorithms and Complexity. Dover, New York (1998)zbMATHGoogle Scholar
  17. 17.
    Sun, Z., Reif, J.: On finding approximate optimal paths in weighted regions. J. Alg. 58, 1–32 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Siu-Wing Cheng
    • 1
  • Man-Kwun Chiu
    • 2
    • 3
  • Jiongxin Jin
    • 4
  • Antoine Vigneron
    • 5
  1. 1.Department of Computer Science and EngineeringHKUSTHong KongHong Kong
  2. 2.National Institute of Informatics (NII)TokyoJapan
  3. 3.JST, ERATOKawarabayashi Large Graph ProjectTokyoJapan
  4. 4.Google Inc.SeattleUSA
  5. 5.Visual Computing CenterKing Abdullah University of Science and Technology (KAUST)ThuwalSaudi Arabia

Personalised recommendations