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How to Morph a Tree on a Small Grid

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Abstract

In this paper we study planar morphs between planar straight-line grid drawings of trees. A morph consists of a sequence of morphing steps, where in a morphing step vertices move along straight-line trajectories at constant speed. We show how to construct planar morphs that simultaneously achieve a reduced number of morphing steps and a polynomially-bounded resolution. We assume that both the initial and final drawings lie on the grid and we ensure that each morphing step produces a grid drawing; further, we consider both upward drawings of rooted trees and drawings of arbitrary trees.

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Notes

  1. When applying Lemma 4.4 in order to transform a canonical drawing of a subtree of T into another canonical drawing of the same subtree of T, we sometimes informally refer to such a transformation as a “rotation”. However, the transformation obtained by means of Lemma 4.4 is a linear morph, and not a rotation. The reason for using the term “rotation” is that the final (canonical) drawing of the morph is the same as the one that we would get by applying an actual \(90^\circ \)-rotation to the initial (canonical) drawing.

References

  1. Alamdari, S., Angelini, P., Barrera-Cruz, F., Chan, T.M., Da Lozzo, G., Di Battista, G., Frati, F., Haxell, P., Lubiw, A., Patrignani, M., Roselli, V., Singla, S., Wilkinson, B.T.: How to morph planar graph drawings. SIAM J. Comput. 46(2), 824–852 (2017)

    Article  MathSciNet  Google Scholar 

  2. Alamdari, S., Angelini, P., Chan, T.M., Di Battista, G., Frati, F., Lubiw, A., Patrignani, M., Roselli, V., Singla, S., Wilkinson, B.T.: Morphing planar graph drawings with a polynomial number of steps. In: 24th Annual ACM-SIAM Symposium on Discrete Algorithms (New Orleans 2013), pp. 1656–1667. SIAM, Philadelphia (2012)

  3. Alexa, M., Cohen-Or, D., Levin, D.: As-rigid-as-possible shape interpolation. In: 27th International Conference on Computer Graphics and Interactive Techniques (New Orleans 2000), pp. 157–164. ACM, New York (2000)

  4. Angelini, P., Da Lozzo, G., Di Battista, G., Frati, F., Patrignani, M., Roselli, V.: Morphing planar graph drawings optimally. In: 41st International Colloquium on Automata, Languages, and Programming (Copenhagen 2014). Part I. Lecture Notes in Comput. Sci., vol. 8572, pp. 126–137. Springer, Heidelberg (2014)

  5. Angelini, P., Da Lozzo, G., Frati, F., Lubiw, A., Patrignani, M., Roselli, V.: Optimal morphs of convex drawings. In: 31st International Symposium on Computational Geometry (Eindhoven 2015). Leibniz Int. Proc. Inform., vol. 34, pp. 126–140. Leibniz-Zent. Inform., Wadern (2015)

  6. Angelini, P., Frati, F., Patrignani, M., Roselli, V.: Morphing planar graph drawings efficiently. In: 21st International Symposium on Graph Drawing (Bordeaux 2013). Lecture Notes in Comput. Sci., vol. 8242, pp. 49–60. Springer, Cham (2013)

  7. Arseneva, E., Bose, P., Cano, P., D’Angelo, A., Dujmović, V., Frati, F., Langerman, S., Tappini, A.: Pole dancing: 3D morphs for tree drawings. J. Graph Algorithms Appl. 23(3), 579–602 (2019)

    Article  MathSciNet  Google Scholar 

  8. Barrera-Cruz, F., Haxell, P., Lubiw, A.: Morphing Schnyder drawings of planar triangulations. Discrete Comput. Geom. 61(1), 161–184 (2019)

    Article  MathSciNet  Google Scholar 

  9. Beier, T., Neely, S.: Feature-based image metamorphosis. In: 19th Annual Conference on Computer Graphics and Interactive Techniques (Chicago 1992), pp. 35–42. ACM, New York (1992)

  10. Biedl, T., Lubiw, A., Petrick, M., Spriggs, M.: Morphing orthogonal planar graph drawings. ACM Trans. Algorithms 9(4), # 29 (2013)

  11. Cairns, S.S.: Deformations of plane rectilinear complexes. Am. Math. Mon. 51(5), 247–252 (1944)

    Article  MathSciNet  Google Scholar 

  12. Carmel, E., Cohen-Or, D.: Warp-guided object-space morphing. Vis. Comput. 13(9–10), 465–478 (1998)

    Article  Google Scholar 

  13. Chambers, E.W., Eppstein, D., Goodrich, M.T., Löffler, M.: Drawing graphs in the plane with a prescribed outer face and polynomial area. J. Graph Algorithms Appl. 16(2), 243–259 (2012)

    Article  MathSciNet  Google Scholar 

  14. Da Lozzo, G., Di Battista, G., Frati, F., Patrignani, M., Roselli, V.: Upward planar morphs. Algorithmica 82(10), 2985–3017 (2020)

    Article  MathSciNet  Google Scholar 

  15. Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing. Algorithms for the Visualization of Graphs. Prentice-Hall, Upper Saddle River (1999)

    MATH  Google Scholar 

  16. Floater, M.S.: Mean value coordinates. Comput. Aided Geom. Des. 20(1), 19–27 (2003)

    Article  MathSciNet  Google Scholar 

  17. Floater, M.S., Gotsman, C.: How to morph tilings injectively. J. Comput. Appl. Math. 101(1–2), 117–129 (1999)

    Article  MathSciNet  Google Scholar 

  18. Friedrich, C., Eades, P.: Graph drawing in motion. J. Graph Algorithms Appl. 6(3), 353–370 (2002)

    Article  MathSciNet  Google Scholar 

  19. Fujimura, K., Makarov, M.: Foldover-free image warping. Graph. Models Image Process. 60(2), 100–111 (1998)

    Article  Google Scholar 

  20. van Goethem, A., Verbeek, K.: Optimal morphs of planar orthogonal drawings. In: 34th International Symposium on Computational Geometry (Budapest 2018). Leibniz Int. Proc. Inform., vol. 99, # 42. Leibniz-Zent. Inform., Wadern (2018)

  21. Gomes, J., Darsa, L., Costa, B., Velho, L.: Warping and Morphing of Graphical Objects. Morgan Kaufmann, San Francisco (1999)

    Google Scholar 

  22. Sederberg, T.W., Gao, P., Wang, G., Mu, H.: 2-D shape blending: an intrinsic solution to the vertex path problem. In: 20th International Conference on Computer Graphics and Interactive Techniques (Anaheim 1993), pp. 15–18. ACM, New York (1993)

  23. Sederberg, T.W., Greenwood, E.: A physically based approach to 2-D shape blending. ACM SIGGRAPH Comput. Graph. 26(2), 25–34 (1992)

    Article  Google Scholar 

  24. Shapira, M., Rappoport, A.: Shape blending using the star-skeleton representation. IEEE Comput. Graph. Appl. 15(2), 44–50 (1995)

    Article  Google Scholar 

  25. Surazhsky, V., Gotsman, C.: Controllable morphing of compatible planar triangulations. ACM Trans. Graph. 20(4), 203–231 (2001)

    Article  Google Scholar 

  26. Tal, A., Elber, G.: Image morphing with feature preserving texture. Comput. Graph. Forum 18(3), 339–348 (1999)

    Article  Google Scholar 

  27. Thomassen, C.: Deformations of plane graphs. J. Comb. Theory Ser. B 34(3), 244–257 (1983)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Sunnyvale, USA

    Fidel Barrera-Cruz

  2. Roma Tre University, Rome, Italy

    Manuel Borrazzo, Giordano Da Lozzo, Giuseppe Di Battista, Fabrizio Frati, Maurizio Patrignani & Vincenzo Roselli

Authors
  1. Fidel Barrera-Cruz
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  2. Manuel Borrazzo
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  3. Giordano Da Lozzo
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  4. Giuseppe Di Battista
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  5. Fabrizio Frati
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  6. Maurizio Patrignani
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  7. Vincenzo Roselli
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Corresponding author

Correspondence to Fabrizio Frati.

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This research was supported in part by MIUR Project “AHeAD” under PRIN 20174LF3T8 and by H2020-MSCA-RISE project 734922—“CONNECT”

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Barrera-Cruz, F., Borrazzo, M., Da Lozzo, G. et al. How to Morph a Tree on a Small Grid. Discrete Comput Geom 67, 743–786 (2022). https://doi.org/10.1007/s00454-021-00363-8

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  • DOI: https://doi.org/10.1007/s00454-021-00363-8

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