Essential Constraints of Edge-Constrained Proximity Graphs

  • Prosenjit Bose
  • Jean-Lou De Carufel
  • Alina Shaikhet
  • Michiel Smid
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9843)

Abstract

Given a plane forest \(F = (V, E)\) of \(|V| = n\) points, we find the minimum set \(S \subseteq E\) of edges such that the edge-constrained minimum spanning tree over the set V of vertices and the set S of constraints contains F. We present an \(O(n \log n)\)-time algorithm that solves this problem. We generalize this to other proximity graphs in the constraint setting, such as the relative neighbourhood graph, Gabriel graph, \(\beta \)-skeleton and Delaunay triangulation.

We present an algorithm that identifies the minimum set \(S\subseteq E\) of edges of a given plane graph \(I=(V,E)\) such that \(I \subseteq CG_\beta (V, S)\) for \(1 \le \beta \le 2\), where \(CG_\beta (V, S)\) is the constraint \(\beta \)-skeleton over the set V of vertices and the set S of constraints. The running time of our algorithm is O(n), provided that the constrained Delaunay triangulation of I is given.

Keywords

Proximity graphs Constraints Visibility MST Delaunay \(\beta \)-skeletons 

References

  1. 1.
    Chew, L.P.: Constrained delaunay triangulations. Algorithmica 4(1), 97–108 (1989)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Chin, F., Wang, C.A.: Finding the constrained Delaunay triangulation and constrained Voronoi diagram of a simple polygon in linear time. SIAM J. Comput. 28(2), 471–486 (1998)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Devillers, O., Estkowski, R., Gandoin, P.-M., Hurtado, F., Ramos, P.A., Sacristán, V.: Minimal set of constraints for 2D constrained Delaunay reconstruction. Int. J. Comput. Geometry Appl. 13(5), 391–398 (2003)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Gabriel, K.R., Sokal, R.R.: A new statistical approach to geographic variation analysis. Syst. Zool. 18(3), 259–278 (1969)CrossRefGoogle Scholar
  5. 5.
    Jaromczyk, J.W., Kowaluk, M.: A note on relative neighborhood graphs. In: SoCG, pp. 233–241 (1987)Google Scholar
  6. 6.
    Jaromczyk, J.W., Kowaluk, M., Yao, F.: An optimal algorithm for constructing \(\beta \)-skeletons in \(l_p\) metric. Manuscript (1989)Google Scholar
  7. 7.
    Kirkpatrick, D.G., Radke, J.D.: A framework for computational morphology. In: Computational Geometry, vol. 2, pp. 217–248. Machine Intelligence and Pattern Recognition, North-Holland (1985)Google Scholar
  8. 8.
    Lee, D.T., Lin, A.K.: Generalized Dalaunay triangualtion for planar graphs. Discrete Comput. Geometry 1, 201–217 (1986)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Sleator, D.D., Tarjan, R.E.: A data structure for dynamic trees. In: STOC, pp. 114–122. ACM (1981)Google Scholar
  10. 10.
    Toussaint, G.T.: The relative neighbourhood graph of a finite planar set. Pattern Recogn. 12(4), 261–268 (1980)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Prosenjit Bose
    • 1
  • Jean-Lou De Carufel
    • 2
  • Alina Shaikhet
    • 1
  • Michiel Smid
    • 1
  1. 1.School of Computer ScienceCarleton UniversityOttawaCanada
  2. 2.School of Electrical Engineering and Computer ScienceUniversity of OttawaOttawaCanada

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