Colored Spanning Graphs for Set Visualization
We study an algorithmic problem that is motivated by ink minimization for sparse set visualizations. Our input is a set of points in the plane which are either blue, red, or purple. Blue points belong exclusively to the blue set, red points belong exclusively to the red set, and purple points belong to both sets. A red-blue-purple spanning graph (RBP spanning graph) is a set of edges connecting the points such that the subgraph induced by the red and purple points is connected, and the subgraph induced by the blue and purple points is connected.
We study the geometric properties of minimum RBP spanning graphs and the algorithmic problems associated with computing them. Specifically, we show that the general problem is NP-hard. Hence we give an \((\frac 12\rho+1)\)-approximation, where ρ is the Steiner ratio. We also present efficient exact solutions if the points are located on a line or a circle. Finally we consider extensions to more than two sets.
Unable to display preview. Download preview PDF.
- 3.Alper, B., Riche, N., Ramos, G., Czerwinski, M.: Design study of LineSets, a novel set visualization technique. IEEE TVCG 17(12), 2259–2267 (2011)Google Scholar
- 9.Collins, C., Penn, G., Carpendale, S.: Bubble Sets: Revealing set relations with isocontours over existing visualizations. IEEE TVCG 15(6), 1009–1016 (2009)Google Scholar
- 11.Edwards, A.W.F.: Cogwheels of the mind. John Hopkins University Press (2004)Google Scholar
- 13.Henry Riche, N., Dwyer, T.: Untangling Euler diagrams. IEEE TVCG 16(6), 1090–1099 (2010)Google Scholar
- 14.Kaneko, A., Kano, M.: Discrete geometry on red and blue points in the plane – a survey. In: Discrete and Comp. Geometry, The Goodman-Pollack Festschrift, pp. 551–570 (2003)Google Scholar
- 15.Meulemans, W., Henry Riche, N., Speckmann, B., Alper, B., Dwyer, T.: KelpFusion: a hybrid set visualization technique. In: IEEE TVCG (to appear, 2013)Google Scholar
- 16.Mitchell, J.S.B.: Geometric shortest paths and network optimization. In: Handbook of Computational Geometry, pp. 633–701 (1998)Google Scholar
- 20.Simonetto, P., Auber, D.: Visualise undrawable Euler diagrams. In: Proc. 12th Conf. on Information Visualisation, pp. 594–599 (2008)Google Scholar
- 21.Stapleton, G., Rodgers, P., Howse, J., Zhang, L.: Inductively generating Euler diagrams. IEEE TVCG 17(1), 88–100 (2011)Google Scholar
- 23.Tufte, E.R.: The Visual Display of Quantitative Information. Graphics Press (1983)Google Scholar