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Tree Transformation through Vertex Contraction with Application to Skeletons

  • Arseny Smirnov
  • Kira Vyatkina
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6784)

Abstract

We start with the problem of verifying ε-equivalence between the medial and a linear axes for a simple polygon, and restate it as a problem of transforming a tree with labeled leaved into another one through contraction of inner vertices in presence of certain restrictions. We demonstrate that a possibility to contract non-adjacent vertices is sometimes crucial for the initial task.

Next, we provide a linear algorithm for solving a relaxed problem on trees, when any two inner vertices may be glued together. We further show that if a required transformation of a given tree can be performed, then it can also be accomplished in such a way that after each contraction, the obtained intermediate graph is a tree, and the respective sequence of merges can be retrieved in linear time.

Keywords

Adjacent Vertex Medial Axis Simple Polygon Matching Coloring Geometric Graph 
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.
    Aichholzer, O., Aurenhammer, F., Alberts, D., Gärtner, B.: A novel type of skeleton for polygons. The Journal of Universal Computer Science 1, 752–761 (1995)zbMATHMathSciNetGoogle Scholar
  2. 2.
    Aichholzer, O., Aurenhammer, F.: Straight skeletons for general polygonal figures. In: Cai, J.-Y., Wong, C.K. (eds.) COCOON 1996. LNCS, vol. 1090, pp. 117–126. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  3. 3.
    Blum, H.: A transformation for extracting new descriptors of shape. In: Dunn, W.W. (ed.) Proc. Symp. Models for the Perception of Speech and Visual Form, pp. 362–380. MIT Press, Cambridge (1967)Google Scholar
  4. 4.
    Tǎnase, M.: Shape Decomposition and Retrieval. Ph.D. Thesis, Utrecht University  (2005)Google Scholar
  5. 5.
    Tǎnase, M., Veltkamp, R.C.: Straight skeleton approximating the medial axis. In: Proc. 12th Ann. European Symp. on Algorithms, pp. 809–821 (2004)Google Scholar
  6. 6.
    Trofimov, V., Vyatkina, K.: Linear axis for general polygons: properties and computation. In: Gervasi, O., Gavrilova, M.L. (eds.) ICCSA 2007, Part I. LNCS, vol. 4705, pp. 122–135. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  7. 7.
    Vyatkina, K.: Linear axis for planar straight line graphs. In: Downey, R., Manyem, P. (eds.) Proc. CATS 2009, CRPIT 94, pp. 137–150 (2009)Google Scholar
  8. 8.
    Warnow, T.: Tree compatibility and inferring evolutionary history. Journal of Algorithms 16, 388–407 (1994)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Arseny Smirnov
    • 1
  • Kira Vyatkina
    • 1
  1. 1.Dept. of Mathematics and MechanicsSaint Petersburg State UniversityStary PeterhofRussia

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