, Volume 76, Issue 1, pp 225–258 | Cite as

Multi-sided Boundary Labeling

  • Philipp Kindermann
  • Benjamin Niedermann
  • Ignaz Rutter
  • Marcus Schaefer
  • André Schulz
  • Alexander Wolff


In the Boundary Labeling problem, we are given a set of n points, referred to as sites, inside an axis-parallel rectangle R, and a set of n pairwise disjoint rectangular labels that are attached to R from the outside. The task is to connect the sites to the labels by non-intersecting rectilinear paths, so-called leaders, with at most one bend. In this paper, we study the Multi-Sided Boundary Labeling problem, with labels lying on at least two sides of the enclosing rectangle. We present a polynomial-time algorithm that computes a crossing-free leader layout if one exists. So far, such an algorithm has only been known for the cases in which labels lie on one side or on two opposite sides of R (here a crossing-free solution always exists). The case where labels may lie on adjacent sides is more difficult. We present efficient algorithms for testing the existence of a crossing-free leader layout that labels all sites and also for maximizing the number of labeled sites in a crossing-free leader layout. For two-sided boundary labeling with adjacent sides, we further show how to minimize the total leader length in a crossing-free layout.


Computational geometry Boundary labeling Dynamic program 


  1. 1.
    Agarwal, P.K., Efrat, A., Sharir, M.: Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. SIAM J. Comput. 29(3), 912–953 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bastert, O., Fekete, S.P.: Geometrische Verdrahtungsprobleme. Technical Report 96–247, Universität zu Köln (1996)Google Scholar
  3. 3.
    Bekos, M.A., Cornelsen, S., Fink, M., Hong, S., Kaufmann, M., Nöllenburg, M., Rutter, I., Symvonis, A.: Many-to-one boundary labeling with backbones. In: Wismath, S., Wolff, A. (eds.) Proceedings of 21st International Symposium on Graph Drawing (GD’13), Volume 8242 of Lecture Notes in Computer Science, pp. 244–255. Springer, Berlin (2013)Google Scholar
  4. 4.
    Bekos, M.A., Kaufmann, M., Nöllenburg, M., Symvonis, A.: Boundary labeling with octilinear leaders. Algorithmica 57(3), 436–461 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bekos, M.A., Kaufmann, M., Potika, K., Symvonis, A.: Area-feature boundary labeling. Comput. J. 53(6), 827–841 (2010)CrossRefGoogle Scholar
  6. 6.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Benkert, M., Haverkort, H.J., Kroll, M., Nöllenburg, M.: Algorithms for multi-criteria boundary labeling. J. Graph Algorithms Appl. 13(3), 289–317 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chan, T.M., Hoffmann, H.-F., Kiazyk, S., Lubiw, A.: Minimum length embedding of planar graphs at fixed vertex locations. In: Wismath, S.K., Wolff, A. (eds.) Proceedings of the 21st International Symposium on Graph Drawing (GD’13) volume 8242 of Lecture Notes in Computer Science, pp. 376–387. Springer, Berlin (2013)Google Scholar
  9. 9.
    Chazelle, B., 36 co-authors.: The computational geometry impact task force report. In: Chazelle, B. Goodman, J.E., Pollack, R., (eds.) Advances in Discrete and Computational Geometry, vol. 223, pp. 407–463. American Mathematical Society, Providence (1999)Google Scholar
  10. 10.
    Fink, M., Haunert, J.-H., Schulz, A., Spoerhase, J., Wolff, A.: Algorithms for labeling focus regions. IEEE Trans. Visual. Comput. Graphics 18(12), 2583–2592 (2012)CrossRefGoogle Scholar
  11. 11.
    Freeman, H., Marrinan, S., Chitalia, H.: Automated labeling of soil survey maps. In: Proceedings of ASPRS-ACSM Annual Convention, Baltimore, Vol. 1, pp. 51–59 (1996)Google Scholar
  12. 12.
    Gemsa, A., Haunert, J.-H., Nöllenburg, M.: Boundary-labeling algorithms for panorama images. In: Proceedings of the 19th ACM SIGSPATIAL International Conference on Advanced Geographic Information Systems (ACM-GIS’11), pp. 289–298 (2011)Google Scholar
  13. 13.
    Gritzmann, P., Mohar, B., Pach, J., Pollack, R.: Embedding a planar triangulation with vertices at specified positions. Am. Math. Mon. 98, 165–166 (1991)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Hirschberg, D.S.: A linear space algorithm for computing maximal common subsequences. Commun. ACM 18(6), 341–343 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Katz, B., Krug, M., Rutter, I., Wolff, A.: Manhattan-geodesic embedding of planar graphs. In: Eppstein, D., Gansner, E.R. (eds) Proceedings of the 17th International Symposium Graph Drawing (GD’09), Volume 5849 of Lecture Notes in Computer Science, pp. 207–218. Springer, Berlin (2010)Google Scholar
  16. 16.
    Liebling, T.M., Margot, F., Müller, D., Prodon, A., Stauffer, L.: Disjoint paths in the plane. ORSA J. Comput. 7(1), 84–88 (1995)CrossRefzbMATHGoogle Scholar
  17. 17.
    Lin, C.-C.: Crossing-free many-to-one boundary labeling with hyperleaders. In: Proceedings of the IEEE Pacific Visualization Symposium(PacificVis’10), pp. 185–192 (2010)Google Scholar
  18. 18.
    Lin, C.-C., Kao, H.-J., Yen, H.-C.: Many-to-one boundary labeling. J. Graph Algorithms Appl. 12(3), 319–356 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Morrison, J.L.: Computer technology and cartographic change. In: Taylor, D. (ed.) The Computer in Contemporary Cartography. Johns Hopkins University Press, Baltimore (1980)Google Scholar
  20. 20.
    Nöllenburg, M., Polishchuk, V., Sysikaski, M.: Dynamic one-sided boundary labeling. In: Proceedings of the 18th ACM SIGSPATIAL International Symposium on Advances in Geographic Information Systems (ACM-GIS’10), pp. 310–319 (2010)Google Scholar
  21. 21.
    Raghavan, R., Cohoon, J., Sahni, S.: Single bend wiring. J. Algorithms 7(2), 232–257 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    van Kreveld, M., Strijk, T., Wolff, A.: Point labeling with sliding labels. Comput. Geom. Theory Appl. 13, 21–47 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Zoraster, S.: Practical results using simulated annealing for point feature label placement. Cartogr. GIS 24(4), 228–238 (1997)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Philipp Kindermann
    • 1
  • Benjamin Niedermann
    • 2
  • Ignaz Rutter
    • 2
  • Marcus Schaefer
    • 3
  • André Schulz
    • 4
  • Alexander Wolff
    • 1
  1. 1.Lehrstuhl für Informatik IUniversität WürzburgWürzburgGermany
  2. 2.Fakultät für InformatikKarlsruher Institut für Technologie (KIT)KarlsruheGermany
  3. 3.College of Computing and Digital MediaDePaul UniversityChicagoUSA
  4. 4.Institut für Mathematische Logik und GrundlagenforschungUniversität MünsterMünsterGermany

Personalised recommendations