Advertisement

Greedy Rectilinear Drawings

  • Patrizio Angelini
  • Michael A. Bekos
  • Walter Didimo
  • Luca Grilli
  • Philipp KindermannEmail author
  • Tamara Mchedlidze
  • Roman Prutkin
  • Antonios Symvonis
  • Alessandra Tappini
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11282)

Abstract

A drawing of a graph is greedy if for each ordered pair of vertices u and v, there is a path from u to v such that the Euclidean distance to v decreases monotonically at every vertex of the path. The existence of greedy drawings has been widely studied under different topological and geometric constraints, such as planarity, face convexity, and drawing succinctness. We introduce greedy rectilinear drawings, in which each edge is either a horizontal or a vertical segment. These drawings have several properties that improve human readability and support network routing. We address the problem of testing whether a planar rectilinear representation, i.e., a plane graph with specified vertex angles, admits vertex coordinates that define a greedy drawing. We provide a characterization, a linear-time testing algorithm, and a full generative scheme for universal greedy rectilinear representations, i.e., those for which every drawing is greedy. For general greedy rectilinear representations, we give a combinatorial characterization and, based on it, a polynomial-time testing and drawing algorithm for a meaningful subset of instances.

References

  1. 1.
    Alamdari, S., Chan, T.M., Grant, E., Lubiw, A., Pathak, V.: Self-approaching graphs. In: Didimo, W., Patrignani, M. (eds.) GD 2012. LNCS, vol. 7704, pp. 260–271. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-36763-2_23CrossRefGoogle Scholar
  2. 2.
    Angelini, P., et al.: Greedy rectilinear drawings. Arxiv report 1808.09063 (2018). http://arxiv.org/abs/1808.09063
  3. 3.
    Angelini, P., Di Battista, G., Frati, F.: Succinct greedy drawings do not always exist. Networks 59(3), 267–274 (2012).  https://doi.org/10.1002/net.21449MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Angelini, P., Frati, F., Grilli, L.: An algorithm to construct greedy drawings of triangulations. J. Graph Algorithms Appl. 14(1), 19–51 (2010).  https://doi.org/10.7155/jgaa.00197MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Batini, C., Nardelli, E., Tamassia, R.: A layout algorithm for data flow diagrams. IEEE Trans. Software Eng. 12(4), 538–546 (1986).  https://doi.org/10.1109/TSE.1986.6312901CrossRefGoogle Scholar
  6. 6.
    Bridgeman, S.S., Di Battista, G., Didimo, W., Liotta, G., Tamassia, R., Vismara, L.: Turn-regularity and optimal area drawings of orthogonal representations. Comput. Geom. 16(1), 53–93 (2000).  https://doi.org/10.1016/S0925-7721(99)00054-1MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Da Lozzo, G., D’Angelo, A., Frati, F.: On planar greedy drawings of 3-connected planar graphs. In: Aronov, B., Katz, M.J. (eds.) Proceedings 33rd International Symposium on Computational Geometry (SoCG 2017). LIPIcs, vol. 77, pp. 33:1–33:16 (2017).  https://doi.org/10.4230/LIPIcs.SoCG.2017.33
  8. 8.
    Dhandapani, R.: Greedy drawings of triangulations. Discrete Comput. Geom. 43(2), 375–392 (2010).  https://doi.org/10.1007/s00454-009-9235-6MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall, Englewood Cliffs (1999)zbMATHGoogle Scholar
  10. 10.
    Duncan, C.A., Goodrich, M.T.: Planar orthogonal and polyline drawing algorithms. In: Tamassia, R. (ed.) Handbook on Graph Drawing and Visualization. Chapman and Hall/CRC (2013). https://cs.brown.edu/~rt/gdhandbook/
  11. 11.
    Eppstein, D., Goodrich, M.T.: Succinct greedy geometric routing using hyperbolic geometry. IEEE Trans. Comput. 60(11), 1571–1580 (2011).  https://doi.org/10.1109/TC.2010.257MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Goodrich, M.T., Strash, D.: Succinct greedy geometric routing in the euclidean plane. In: Dong, Y., Du, D.-Z., Ibarra, O. (eds.) ISAAC 2009. LNCS, vol. 5878, pp. 781–791. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-642-10631-6_79CrossRefGoogle Scholar
  13. 13.
    He, X., Zhang, H.: On succinct convex greedy drawing of 3-connected plane graphs. In: Randall, D. (ed.) Proceedings 22nd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2011), pp. 1477–1486. SIAM (2011).  https://doi.org/10.1137/1.9781611973082.115CrossRefGoogle Scholar
  14. 14.
    He, X., Zhang, H.: On succinct greedy drawings of plane triangulations and 3-connected plane graphs. Algorithmica 68(2), 531–544 (2014).  https://doi.org/10.1007/s00453-012-9682-yMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Icking, C., Klein, R., Langetepe, E.: Self-approaching curves. Math. Proc. Camb. Phil. Soc. 125, 441–443 (1999).  https://doi.org/10.1017/S0305004198003016MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Leighton, T., Moitra, A.: Some results on greedy embeddings in metric spaces. Discrete Comput. Geom. 44(3), 686–705 (2010).  https://doi.org/10.1007/s00454-009-9227-6MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Leone, P., Samarasinghe, K.: Geographic routing on virtual raw anchor coordinate systems. Theor. Comput. Sci. 621, 1–13 (2016).  https://doi.org/10.1016/j.tcs.2015.12.029MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Nöllenburg, M., Prutkin, R.: Euclidean greedy drawings of trees. In: Bodlaender, H.L., Italiano, G.F. (eds.) ESA 2013. LNCS, vol. 8125, pp. 767–778. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-40450-4_65CrossRefGoogle Scholar
  19. 19.
    Nöllenburg, M., Prutkin, R., Rutter, I.: On self-approaching and increasing-chord drawings of 3-connected planar graphs. J. Comput. Geom. 7(1), 47–69 (2016).  https://doi.org/10.20382/jocg.v7i1a3MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Papadimitriou, C.H., Ratajczak, D.: On a conjecture related to geometric routing. In: Nikoletseas, S.E., Rolim, J.D.P. (eds.) ALGOSENSORS 2004. LNCS, vol. 3121, pp. 9–17. Springer, Heidelberg (2004).  https://doi.org/10.1007/978-3-540-27820-7_3CrossRefGoogle Scholar
  21. 21.
    Papadimitriou, C.H., Ratajczak, D.: On a conjecture related to geometric routing. Theor. Comput. Sci. 344(1), 3–14 (2005).  https://doi.org/10.1016/j.tcs.2005.06.022MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Rao, A., Papadimitriou, C.H., Shenker, S., Stoica, I.: Geographic routing without location information. In: Johnson, D.B., Joseph, A.D., Vaidya, N.H. (eds.) Proceedings of 9th Annual International Conference on Mobile Computing and Networking (MOBICOM 2003), pp. 96–108. ACM (2003).  https://doi.org/10.1145/938985.938996
  23. 23.
    Tamassia, R.: On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput. 16(3), 421–444 (1987).  https://doi.org/10.1137/0216030MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Wang, J., He, X.: Succinct strictly convex greedy drawing of 3-connected plane graphs. Theor. Comput. Sci. 532, 80–90 (2014).  https://doi.org/10.1016/j.tcs.2013.05.024MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Patrizio Angelini
    • 1
  • Michael A. Bekos
    • 1
  • Walter Didimo
    • 2
  • Luca Grilli
    • 2
  • Philipp Kindermann
    • 3
    Email author
  • Tamara Mchedlidze
    • 4
  • Roman Prutkin
    • 4
  • Antonios Symvonis
    • 5
  • Alessandra Tappini
    • 2
  1. 1.Institut für InformatikUniversität TübingenTübingenGermany
  2. 2.Università degli Studi di PerugiaPerugiaItaly
  3. 3.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  4. 4.Institute of Theoretical InformaticsKarlsruhe Institute of TechnologyKarlsruheGermany
  5. 5.School of Applied Mathematical and Physical Sciences, NTUAAthensGreece

Personalised recommendations