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%.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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)
Buchsbaum, A.L., Gansner, E.R., Procopiuc, C.M., Venkatasubramanian, S.: Rectangular layouts and contact graphs. ACM Trans. Algorithms 4(1) (2008)
Chekuri, C., Khanna, S.: A polynomial time approximation scheme for the multiple knapsack problem. SIAM J. Comput. 35(3), 713–728 (2005)
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)
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)
Dumais, S.T.: Latent semantic analysis. Annu. Rev. Inform. Sci. Tech. 38(1), 188–230 (2004)
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)
Eppstein, D., Mumford, E., Speckmann, B., Verbeek, K.: Area-universal and constrained rectangular layouts. SIAM J. Comput. 41(3), 537–564 (2012)
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)
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)
Gansner, E.R., Hu, Y.: Efficient, proximity-preserving node overlap removal. J. Graph Algortihms Appl. 14(1), 53–74 (2010)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)
Hakimi, S.L., Mitchem, J., Schmeichel, E.F.: Star arboricity of graphs. Discrete Math. 149(1-3), 93–98 (1996)
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)
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)
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)
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)
Nocaj, A., Brandes, U.: Organizing search results with a reference map. IEEE Trans. Vis. Comput. Graphics 18(12), 2546–2555 (2012)
Nöllenburg, M., Prutkin, R., Rutter, I.: Edge-weighted contact representations of planar graphs. J. Graph Algorithms Appl. 17(4), 441–473 (2013)
Petersen, J.: Die Theorie der regulären Graphen. Acta Mathematica 15(1), 193–220 (1891)
Raisz, E.: The rectangular statistical cartogram. Geogr. Review 24(3), 292–296 (1934)
Thomassen, C.: Interval representations of planar graphs. J. Combin. Theory, Ser. B 40(1), 9–20 (1986)
Viégas, F.B., Wattenberg, M., Feinberg, J.: Participatory visualization with Wordle. IEEE Trans. Vis. Comput. Graphics 15(6), 1137–1144 (2009)
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Barth, L. et al. (2014). Semantic Word Cloud Representations: Hardness and Approximation Algorithms. In: Pardo, A., Viola, A. (eds) LATIN 2014: Theoretical Informatics. LATIN 2014. Lecture Notes in Computer Science, vol 8392. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54423-1_45
Download citation
DOI: https://doi.org/10.1007/978-3-642-54423-1_45
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-54422-4
Online ISBN: 978-3-642-54423-1
eBook Packages: Computer ScienceComputer Science (R0)