Semantic Word Cloud Representations: Hardness and Approximation Algorithms

  • Lukas Barth
  • Sara Irina Fabrikant
  • Stephen G. Kobourov
  • Anna Lubiw
  • Martin Nöllenburg
  • Yoshio Okamoto
  • Sergey Pupyrev
  • Claudio Squarcella
  • Torsten Ueckerdt
  • Alexander Wolff
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8392)

Abstract

We study a geometric representation problem, where we are given a set \(\mathcal B\) of axis-aligned rectangles (boxes) with fixed dimensions and a graph with vertex set \(\mathcal B\). The task is to place the rectangles without overlap such that two rectangles touch if the graph contains an edge between them. We call this problem Contact Representation of Word Networks (Crown). It formalizes the geometric problem behind drawing word clouds in which semantically related words are close to each other. Here, we represent words by rectangles and semantic relationships by edges.

We show that Crown is strongly NP-hard even if restricted to trees and weakly NP-hard if restricted to stars. We also consider the optimization problem Max-Crown where each adjacency induces a certain profit and the task is to maximize the sum of the profits. For this problem, we present constant-factor approximations for several graph classes, namely stars, trees, planar graphs, and graphs of bounded degree. Finally, we evaluate the algorithms experimentally and show that our best method improves upon the best existing heuristic by 45%.

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References

  1. 1.
    Barth, L., Fabrikant, S.I., Kobourov, S., Lubiw, A., Nöllenburg, M., Okamoto, Y., Pupyrev, S., Squarcella, C., Ueckerdt, T., Wolff, A.: Semantic word cloud representations: Hardness and approximation algorithms. Arxiv report arxiv.org/abs/1311.4778 (2013)Google Scholar
  2. 2.
    Buchsbaum, A.L., Gansner, E.R., Procopiuc, C.M., Venkatasubramanian, S.: Rectangular layouts and contact graphs. ACM Trans. Algorithms 4(1) (2008)Google Scholar
  3. 3.
    Chekuri, C., Khanna, S.: A polynomial time approximation scheme for the multiple knapsack problem. SIAM J. Comput. 35(3), 713–728 (2005)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Collins, C., Viégas, F.B., Wattenberg, M.: Parallel tag clouds to explore and analyze faceted text corpora. In: Proc. IEEE Symp. Vis. Analytics Sci. Tech., pp. 91–98 (2009)Google Scholar
  5. 5.
    Cui, W., Wu, Y., Liu, S., Wei, F., Zhou, M., Qu, H.: Context-preserving dynamic word cloud visualization. IEEE Comput. Graphics Appl. 30(6), 42–53 (2010)CrossRefGoogle Scholar
  6. 6.
    Dumais, S.T.: Latent semantic analysis. Annu. Rev. Inform. Sci. Tech. 38(1), 188–230 (2004)CrossRefGoogle Scholar
  7. 7.
    Dwyer, T., Marriott, K., Stuckey, P.J.: Fast node overlap removal. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 153–164. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  8. 8.
    Eppstein, D., Mumford, E., Speckmann, B., Verbeek, K.: Area-universal and constrained rectangular layouts. SIAM J. Comput. 41(3), 537–564 (2012)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Felsner, S.: Rectangle and square representations of planar graphs. In: Pach, J. (ed.) Thirty Essays on Geometric Graph Theory, pp. 213–248. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  10. 10.
    Fleischer, L., Goemans, M.X., Mirrokni, V.S., Sviridenko, M.: Tight approximation algorithms for maximum separable assignment problems. Math. Oper. Res. 36(3), 416–431 (2011)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Gansner, E.R., Hu, Y.: Efficient, proximity-preserving node overlap removal. J. Graph Algortihms Appl. 14(1), 53–74 (2010)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)Google Scholar
  13. 13.
    Hakimi, S.L., Mitchem, J., Schmeichel, E.F.: Star arboricity of graphs. Discrete Math. 149(1-3), 93–98 (1996)Google Scholar
  14. 14.
    Koh, K., Lee, B., Kim, B.H., Seo, J.: Maniwordle: Providing flexible control over Wordle. IEEE Trans. Vis. Comput. Graph. 16(6), 1190–1197 (2010)CrossRefGoogle Scholar
  15. 15.
    Lagus, K., Honkela, T., Kaski, S., Kohonen, T.: Self-organizing maps of document collections: A new approach to interactive exploration. In: Simoudis, E., Han, J., Fayyad, U.M. (eds.) KDD 1996, pp. 238–243. AAAI Press (1996)Google Scholar
  16. 16.
    Leung, J.Y.T., Tam, T.W., Wong, C., Young, G.H., Chin, F.Y.: Packing squares into a square. J. Parallel Distrib. Comput. 10(3), 271–275 (1990)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Nguyen, C.T., Shen, J., Hou, M., Sheng, L., Miller, W., Zhang, L.: Approximating the spanning star forest problem and its application to genomic sequence alignment. SIAM J. Comput. 38(3), 946–962 (2008)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Nocaj, A., Brandes, U.: Organizing search results with a reference map. IEEE Trans. Vis. Comput. Graphics 18(12), 2546–2555 (2012)CrossRefGoogle Scholar
  19. 19.
    Nöllenburg, M., Prutkin, R., Rutter, I.: Edge-weighted contact representations of planar graphs. J. Graph Algorithms Appl. 17(4), 441–473 (2013)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Petersen, J.: Die Theorie der regulären Graphen. Acta Mathematica 15(1), 193–220 (1891)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Raisz, E.: The rectangular statistical cartogram. Geogr. Review 24(3), 292–296 (1934)CrossRefGoogle Scholar
  22. 22.
    Thomassen, C.: Interval representations of planar graphs. J. Combin. Theory, Ser. B 40(1), 9–20 (1986)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Viégas, F.B., Wattenberg, M., Feinberg, J.: Participatory visualization with Wordle. IEEE Trans. Vis. Comput. Graphics 15(6), 1137–1144 (2009)CrossRefGoogle Scholar
  24. 24.
    Wu, Y., Provan, T., Wei, F., Liu, S., Ma, K.L.: Semantic-preserving word clouds by seam carving. Comput. Graphics Forum 30(3), 741–750 (2011)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Lukas Barth
    • 1
  • Sara Irina Fabrikant
    • 2
  • Stephen G. Kobourov
    • 3
  • Anna Lubiw
    • 4
  • Martin Nöllenburg
    • 1
  • Yoshio Okamoto
    • 5
  • Sergey Pupyrev
    • 3
    • 9
  • Claudio Squarcella
    • 6
  • Torsten Ueckerdt
    • 7
  • Alexander Wolff
    • 8
  1. 1.Institute of Theoretical InformaticsKarlsruhe Institute of TechnologyGermany
  2. 2.Department of GeographyUniversity of ZurichSwitzerland
  3. 3.Department of Computer ScienceUniversity of ArizonaUSA
  4. 4.David R. Cheriton School of Computer ScienceUniversity of WaterlooCanada
  5. 5.Dept. Comm. Engineering and InformaticsUniversity of Electro-CommunicationsJapan
  6. 6.Dipartimento di IngegneriaRoma Tre UniversityItaly
  7. 7.Department of MathematicsKarlsruhe Institute of TechnologyGermany
  8. 8.Lehrstuhl für Informatik IUniversität WürzburgGermany
  9. 9.Institute of Mathematics and Computer ScienceUral Federal UniversityRussia

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