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Semi-dynamic Connectivity in the Plane

  • Sergio Cabello
  • Michael Kerber
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9214)

Abstract

Motivated by a path planning problem we consider the following procedure. Assume that we have two points s and t in the plane and take \(\mathcal {K}=\emptyset \). At each step we add to \(\mathcal {K}\) a compact convex set that is disjoint from s and t. We must recognize when the union of the sets in \(\mathcal {K}\) separates s and t, at which point the procedure terminates. We show how to add one set to \(\mathcal {K}\) in \(O(1+k\alpha (n))\) amortized time plus the time needed to find all sets of \(\mathcal {K}\) intersecting the newly added set, where n is the cardinality of \(\mathcal {K}\), k is the number of sets in \(\mathcal {K}\) intersecting the newly added set, and \(\alpha (\cdot )\) is the inverse of the Ackermann function.

Keywords

Intersection Graph Adjacency List Path Planning Problem Polygonal Path Polygonal Curve 
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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References

  1. 1.
    Aizawa, K., Tanaka, S., Motomura, K., Kadowaki, R.: Algorithms for connected component labeling based on quadtrees. International Journal of Imaging Systems and Technology 19(2), 158–166 (2009)CrossRefGoogle Scholar
  2. 2.
    Bennett, H., Yap, C.: Amortized analysis of smooth quadtrees in all dimensions. In: Ravi, R., Gørtz, I.L. (eds.) SWAT 2014. LNCS, vol. 8503, pp. 38–49. Springer, Heidelberg (2014). doi:  10.1007/978-3-319-08404-6_4 CrossRefGoogle Scholar
  3. 3.
    de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry: Algorithms and Applications, 2nd edn. Springer (2000)Google Scholar
  4. 4.
    Cabello, S., Giannopoulos, P.: The complexity of separating points in the plane. Algorithmica (to appear). doi:  10.1007/s00453-014-9965-6
  5. 5.
    Cormen, T.H., Leiverson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. MIT Press (2009)Google Scholar
  6. 6.
    Dasgupta, S., Papadimitriou, C.H., Vazirani, U.V.: Algorithms. McGraw-Hill (2008)Google Scholar
  7. 7.
    Edelsbrunner, H., Mücke, E.P.: Simulation of simplicity: a technique to cope with degenerate cases in geometric algorithms. ACM Transactions on Graphics 9(1), 66–104 (1990). doi:  10.1145/77635.77639 CrossRefzbMATHGoogle Scholar
  8. 8.
    Erickson, J.: Algorithms notes: Maintaining disjoint sets (“union-find”) (2015). http://web.engr.illinois.edu/ jeffe/teaching/algorithms/
  9. 9.
    Lavalle, S.M.: Planning Algorithms. Cambridge University Press (2006)Google Scholar
  10. 10.
    Mohar, B., Thomassen, C.: Graphs on Surfaces. Johns Hopkins University Press (2001)Google Scholar
  11. 11.
    Seidel, R., Sharir, M.: Top-down analysis of path compression. SIAM Journal of Computing 34(3), 515–525 (2005). doi:  10.1137/S0097539703439088 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Tarjan, R.E.: Efficiency of a good but not linear set union algorithm. Journal of the ACM 22(2), 215–225 (1975). doi:  10.1145/321879.321884 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Wang, C., Chiang, Y.-J., Yap, C.: On soft predicates in subdivision motion planning. In: Proceedings of the Twenty-ninth Annual Symposium on Computational Geometry, SoCG 2013, pp. 349–358. ACM (2013). doi:  10.1145/2462356.2462386

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Mathematics, IMFM, and Department of Mathematics, FMFUniversity of LjubljanaLjubljanaSlovenia
  2. 2.Max-Planck-Institut für InformatikSaarbrückenGermany

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