Mixed Map Labeling

  • Maarten Löffler
  • Martin Nöllenburg
  • Frank Staals
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9079)


Point feature map labeling is a geometric problem, in which a set of input points must be labeled with a set of disjoint rectangles (the bounding boxes of the label texts). Typically, labeling models either use internal labels, which must touch their feature point, or external (boundary) labels, which are placed on one of the four sides of the input points’ bounding box and which are connected to their feature points by crossing-free leader lines. In this paper we study polynomial-time algorithms for maximizing the number of internal labels in a mixed labeling model that combines internal and external labels. The model requires that all leaders are parallel to a given orientation \(\theta \in [0,2\pi )\), whose value influences the geometric properties and hence the running times of our algorithms.


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  1. 1.
    Agarwal, P.K., van Kreveld, M., Suri, S.: Label placement by maximum independent set in rectangles. Comput. Geom. Theory Appl. 11(3–4), 209–218 (1998)CrossRefzbMATHGoogle Scholar
  2. 2.
    Bekos, M., Kaufmann, M., Nöllenburg, M., Symvonis, A.: Boundary labeling with octilinear leaders. Algorithmica 57, 436–461 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bekos, M.A., Kaufmann, M., Papadopoulos, D., Symvonis, A.: Combining traditional map labeling with boundary labeling. In: Černá, I., Gyimóthy, T., Hromkovič, J., Jefferey, K., Králović, R., Vukolić, M., Wolf, S. (eds.) SOFSEM 2011. LNCS, vol. 6543, pp. 111–122. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  4. 4.
    Bekos, M.A., Kaufmann, M., Symvonis, A.: Efficient labeling of collinear sites. J. Graph Algorithms Appl. 12(3), 357–380 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Bekos, M.A., Kaufmann, M., Symvonis, A., Wolff, A.: Boundary labeling: Models and efficient algorithms for rectangular maps. Comput. Geom. Theory Appl. 36(3), 215–236 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Benkert, M., Haverkort, H., Kroll, M., Nöllenburg, M.: Algorithms for multi-criteria boundary labeling. J. Graph Algorithms and Appl. 13(3), 289–317 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Chalermsook, P., Chuzhoy, J.: Maximum independent set of rectangles. In: Discrete Algorithms (SODA 2009), pp. 892–901 (2009)Google Scholar
  8. 8.
    de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry: Algorithms and Applications, 2nd edn. Springer-Verlag, Berlin, Germany (2000)CrossRefGoogle Scholar
  9. 9.
    Formann, M., Wagner, F.: A packing problem with applications to lettering of maps. In: Computational Geometry (SoCG 1991), pp. 281–288. ACM (1991)Google Scholar
  10. 10.
    Gemsa, A., Haunert, J.-H., Nöllenburg, M.: Boundary-labeling algorithms for panorama images. In: Advances in Geographic Information Systems (SIGSPATIAL GIS 2011), pp. 289–298. ACM (2011)Google Scholar
  11. 11.
    Huang, Z.-D., Poon, S.-H., Lin, C.-C.: Boundary labeling with flexible label positions. In: Pal, S.P., Sadakane, K. (eds.) WALCOM 2014. LNCS, vol. 8344, pp. 44–55. Springer, Heidelberg (2014) CrossRefGoogle Scholar
  12. 12.
    Imhof, E.: Positioning names on maps. The American Cartographer 2(2), 128–144 (1975)CrossRefGoogle Scholar
  13. 13.
    Kaufmann, M.: On map labeling with leaders. In: Albers, S., Alt, H., Näher, S. (eds.) Efficient Algorithms. LNCS, vol. 5760, pp. 290–304. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  14. 14.
    Kindermann, P., Niedermann, B., Rutter, I., Schaefer, M., Schulz, A., Wolff, A.: Two-sided boundary labeling with adjacent sides. In: Dehne, F., Solis-Oba, R., Sack, J.-R. (eds.) WADS 2013. LNCS, vol. 8037, pp. 463–474. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  15. 15.
    Klau, G.W., Mutzel, P.: Optimal labeling of point features in rectangular labeling models. Mathematical Programming 94(2), 435–458 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Löffler, M., Nöllenburg, M.: Shooting bricks with orthogonal laser beams: a first step towards internal/external map labeling. In: Canadian Conf. Computational Geometry (CCCG 2010), pp. 203–206. University of Manitoba (2010)Google Scholar
  17. 17.
    Löffler, M., Nöllenburg, M., Staals, F.: Mixed map labeling. CoRR, abs/1501.06813 (2015)Google Scholar
  18. 18.
    Marks, J., Shieber, S.: The computational complexity of cartographic label placement. Technical report, Harvard University (1991)Google Scholar
  19. 19.
    Mehlhorn, K., Näher, S.: Dynamic fractional cascading. Algorithmica 5(1–4), 215–241 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Nöllenburg, M., Polishchuk, V., Sysikaski, M.: Dynamic one-sided boundary labeling. In: Advances in Geographic Information Systems (SIGSPATIAL GIS 2010), pp. 310–319, November 2010Google Scholar
  21. 21.
    Reimer, A., Rylov, M.: Point-feature lettering of high cartographic quality: a multi-criteria model with practical implementation. In: EuroCG 2014, Ein-Gedi, Israel (2014)Google Scholar
  22. 22.
    van Kreveld, M., Strijk, T., Wolff, A.: Point labeling with sliding labels. Comput. Geom. Theory Appl. 13(1), 21–47 (1999)CrossRefzbMATHGoogle Scholar
  23. 23.
    Wolff, A., Strijk, T.: The map labeling bibliography.

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Maarten Löffler
    • 1
  • Martin Nöllenburg
    • 2
  • Frank Staals
    • 1
  1. 1.Department of Information and Computing SciencesUtrecht UniversityUtrechtThe Netherlands
  2. 2.Institute of Theoretical InformaticsKarlsruhe Institute of Technology (KIT)KarlsruheGermany

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