The Visual Computer

, Volume 28, Issue 10, pp 1015–1026 | Cite as

Generating optimal drawings of physically realizable symbol maps with integer programming

  • Guilherme Kunigami
  • Pedro J. de Rezende
  • Cid C. de Souza
  • Tallys Yunes
Original Article


Proportional symbol maps are a tool often used by cartographers and geoscience professionals to visualize geopositioned data associated with events and demographic statistics, such as earthquakes and population counts. Symbols are placed at specific locations on a map, and their areas are scaled to become proportional to the magnitudes of the data points they represent. We focus specifically on creating physically realizable drawings of symbols—opaque disks, in our case—by maximizing two quality metrics: the total and the minimum length of their visible borders. As these two maximization problems have been proven to be NP-hard, we provide integer programming formulations for their solution, along with decomposition techniques designed to decrease the size of input instances. Our computational experiments, which use real-life data sets, demonstrate the effectiveness of our approach and provide, for the first time, a number of optimal solutions to previously studied instances of this problem.


Visualization Cartography Computational geometry Integer linear programming 



Guilherme Kunigami is supported by CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) grant 830510/1999-0. Pedro J. de Rezende is partially supported by CNPq grants 483177/2009-1, 473867/2010-9, FAPESP (Fundação de Amparo à Pesquisa do Estado de São Paulo) grant 07/52015-0, and a grant from FAEPEX/UNICAMP. Cid C. de Souza is partially supported by CNPq grants 301732/2007-8, 472504/2007-0, 473867/2010-9, and FAPESP grant 07/52015-0.

The authors would like to express their appreciation to the anonymous referees for their careful and thorough reviews and for their comments which contributed to improving the exposition.


  1. 1.
    Cabello, S., Haverkort, H., van Kreveld, M., Speckmann, B.: Algorithmic aspects of proportional symbol maps. Algorithmica 58(3), 543–565 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    CGAL: Computational Geometry Algorithms Library.
  3. 3.
    Fair Isaac Corp: Xpress optimizer reference manual Google Scholar
  4. 4.
    Griffin, T.: The importance of visual contrast for graduated circles. Cartography 19(1), 21–30 (1990) CrossRefGoogle Scholar
  5. 5.
    Kunigami, G., Cano, R.G., de Rezende, P.J., Yunes, T.H., de Souza, C.C.: Proportional symbol maps—benchmark instances (2011).
  6. 6.
    Kunigami, G., de Rezende, P.J., de Souza, C.C., Yunes, T.: Optimizing the layout of proportional symbol maps.
  7. 7.
    Kunigami, G., de Rezende, P.J., de Souza, C.C., Yunes, T.: Determining an optimal visualization of physically realizable symbol maps. In: Lewiner, T., Torres, R. (eds.) Proc. of the 24th Conf. on Graphics, Patterns and Images. IEEE Comp. Soc. Conf. Pub. Serv., pp. 1–8 (2011) Google Scholar
  8. 8.
    Kunigami, G., de Rezende, P.J., de Souza, C.C., Yunes, T.: Optimizing the layout of proportional symbol maps. In: Murgante, B. et al. (eds.) Proceedings of ICCSA 2011. Lecture Notes in Computer Science, vol. 6784, pp. 1–16. Springer, Berlin (2011) Google Scholar
  9. 9.
    Slocum, T.A., McMaster, R.B., Kessler, F.C., Howard, H.H.: Thematic Cartography and Geographic Visualization, 2nd edn. Prentice Hall, New York (2003) Google Scholar
  10. 10.
    Wolsey, L.A.: Integer Programming. Wiley, New York (1998) zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Guilherme Kunigami
    • 1
  • Pedro J. de Rezende
    • 1
  • Cid C. de Souza
    • 1
  • Tallys Yunes
    • 2
  1. 1.Institute of ComputingUniversity of CampinasCampinasBrazil
  2. 2.Management Science DepartmentUniversity of MiamiCoral GablesUSA

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