, Volume 58, Issue 3, pp 543–565 | Cite as

Algorithmic Aspects of Proportional Symbol Maps

  • Sergio Cabello
  • Herman Haverkort
  • Marc van Kreveld
  • Bettina Speckmann
Open Access


Proportional symbol maps visualize numerical data associated with point locations by placing a scaled symbol—typically an opaque disk or square—at the corresponding point on a map. The area of each symbol is proportional to the numerical value associated with its location. Every visually meaningful proportional symbol map will contain at least some overlapping symbols. These need to be drawn in such a way that the user can still judge their relative sizes accurately.

We identify two types of suitable drawings: physically realizable drawings and stacking drawings. For these we study the following two problems: Max-Min—maximize the minimum visible boundary length of each symbol—and Max-Total—maximize the total visible boundary length over all symbols. We show that both problems are NP-hard for physically realizable drawings. Max-Min can be solved in O(n 2log n) time for stacking drawings, which can be improved to O(nlog n) time when the input has certain properties. We also implemented several methods to compute stacking drawings: our solution to the Max-Min problem performs best on the data sets considered.


Geometric algorithms NP-hardness Cartography 


  1. 1.
    Agarwal, P.K., Suri, S.: Surface approximation and geometric partitions. SIAM J. Comput. 27, 1016–1035 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Cabello, S., Demaine, E.D., Rote, G.: Planar embeddings of graphs with specified edge lengths. J. Graph Algorithms Appl. 11(1), 259–276 (2007) zbMATHMathSciNetGoogle Scholar
  3. 3.
    de Berg, M., Cheong, O., van Kreveld, M., Overmars, M.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer, Berlin (2008) zbMATHGoogle Scholar
  4. 4.
    Demaine, E.D., Mitchell, J.S.B., O’Rourke, J.: The open problems project. Problem 33.
  5. 5.
    Dent, B.: Cartography—Thematic Map Design, 5th edn. McGraw-Hill, New York (1999) Google Scholar
  6. 6.
    Fortnow, L.: Computational Complexity Blog. Post of Friday, February 14, 2003.
  7. 7.
    Fowler, R.J., Paterson, M.S., Tanimoto, S.L.: Optimal packing and covering in the plane are NP-complete. Inf. Process. Lett. 12(3), 133–137 (1981) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Giora, Y., Kaplan, H.: Optimal dynamic vertical ray shooting in rectilinear planar subdivisions. In: Proceedings of the 18th ACM-SIAM Symposium on Discrete Algorithms, pp. 19–28 (2007) Google Scholar
  9. 9.
    Griffin, T.: The importance of visual contrast for graduated circles. Cartography 19(1), 21–30 (1990) Google Scholar
  10. 10.
    Groop, R.E., Cole, D.: Overlapping graduated circles: Magnitude estimation and method of portrayal. Can. Cartogr. 15(2), 114–122 (1978) Google Scholar
  11. 11.
    Imai, H., Asano, T.: Dynamic orthogonal segment intersection search. J. Algorithms 8, 1–18 (1987) zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kedem, K., Livne, R., Pach, J., Sharir, M.: On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles. Discrete Comput. Geom. 1, 59–71 (1986) zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Knuth, D.E., Raghunathan, A.: The problem of compatible representatives. SIAM J. Discrete Math. 5, 422–427 (1992) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Lichtenstein, D.: Planar formulae and their uses. SIAM J. Comput. 11(2), 329–343 (1982) zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Mehlhorn, K., Näher, S.: Dynamic fractional cascading. Algorithmica 5, 215–241 (1990) zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    NOAA Satellite and Information Service. National geophysical data center, 2005.
  17. 17.
    Queensland University Advanced Centre for Earthquake Studies.
  18. 18.
    Sharir, M.: On k-sets in arrangements of curves and surfaces. Discrete Comput. Geom. 6, 593–613 (1991) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    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

Copyright information

© The Author(s) 2009

Authors and Affiliations

  • Sergio Cabello
    • 1
  • Herman Haverkort
    • 2
  • Marc van Kreveld
    • 3
  • Bettina Speckmann
    • 2
  1. 1.Department of MathematicsInstitute for Mathematics, Physics and MechanicsLjubljanaSlovenia
  2. 2.Department of Mathematics and Computer ScienceTU EindhovenEindhovenThe Netherlands
  3. 3.Department of Information and Computing SciencesUtrecht UniversityUtrechtThe Netherlands

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