Abstract
Bipartite graphs are commonly used to visualize objects and their features. An object may possess several features and several objects may share a common feature. The standard visualization of bipartite graphs, with objects and features on two (say horizontal) parallel lines at integer coordinates and edges drawn as line segments, can often be difficult to work with. A common task in visualization of such graphs is to consider one object and all its features. This naturally defines a drawing window, defined as the smallest interval that contains the x-coordinates of the object and all its features. We show that if both objects and features can be reordered, minimizing the average window size is NP-hard. However, if the features are fixed, then we provide an efficient polynomial time algorithm for arranging the objects, so as to minimize the average window size. Finally, we introduce a different way of visualizing the bipartite graph, by placing the nodes of the two parts on two concentric circles. For this setting we also show NP-hardness for the general case and a polynomial time algorithm when the features are fixed.
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References
Ahmed, R., et al.: Splitting vertices in 2-layer graph drawings. IEEE Comput. Graph. Appl. (2023)
Baumann, J., Pfretzschner, M., Rutter, I.: Parameterized complexity of vertex splitting to pathwidth at most 1. arXiv preprint arXiv:2302.14725 (2023)
Bekos, M.A., et al.: On the 2-layer window width minimization problem. In: Gasieniec, L. (ed.) SOFSEM 2023. LNCS, vol. 13878, pp. 209–221. Springer, Cham (2023)
Bhat, K.V.S.: An \({O}(n^{2.5} \log _2 n)\) time algorithm for the bottleneck assignment problem. unpublished. AT &T Bell Laboratories, Napiendle, IL (1984)
Buchin, K., et al.: Drawing (complete) binary tanglegrams. Algorithmica 62(1–2), 309–332 (2012). https://doi.org/10.1007/s00453-010-9456-3
Chen, L., Kyng, R., Liu, Y.P., Peng, R., Gutenberg, M.P., Sachdeva, S.: Maximum flow and minimum-cost flow in almost-linear time. In 2022 IEEE 63rd Annual Symposium on Foundations of Computer Science (FOCS), pp. 612–623. IEEE (2022)
Battista, G.D., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall, Hoboken (1999)
Eades, P., de Mendonça N, C.F.X.: Vertex splitting and tension-free layout. In: Brandenburg, F.J. (ed.) GD 1995. LNCS, vol. 1027, pp. 202–211. Springer, Heidelberg (1996). https://doi.org/10.1007/BFb0021804
Eades, P., Wormald, N.C.: Edge crossings in drawings of bipartite graphs. Algorithmica 11(4), 379–403 (1994)
Fernau, H., Kaufmann, M., Poths, M.: Comparing trees via crossing minimization. J. Comput. Syst. Sci. 76(7), 593–608 (2010). https://doi.org/10.1016/j.jcss.2009.10.014
Gabow, H.N., Tarjan, R.E.: Algorithms for two bottleneck optimization problems. J. Algorithms 9, 411–417 (1988)
Garey, M.R., Johnson, D.S., Stockmeyer, L.: Some simplified NP-complete graph problems. Theoret. Comput. Sci. 1(3), 237–267 (1976)
Glover, F.: Maximum matching in convex bipartite graphs. Naval Res. Logistic Q. 14, 313–316 (1967)
Kaufmann, M., Wagner, D.: Drawing Graphs, Methods and Models, vol. 2025. Springer, Cham (2001). https://doi.org/10.1007/3-540-44969-8
Liang, Y.D., Blum, N.: Circular convex bipartite graphs: maximum matching and Hamiltonian circuits. Inf. Process. Lett. 56(4), 215–219 (1995)
Nöllenburg, M., Sorge, M., Terziadis, S., Villedieu, A., Wu, H.Y., Wulms, J.: Planarizing graphs and their drawings by vertex splitting. In: Angelini, P., von Hanxleden, R. (eds.) GD 2022. LNCS, vol. 13764, pp. 232–246. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-22203-0_17
Steiner, G., Yeomans, J.S.: A linear time algorithm for determining maximum matchings in convex, bipartite graphs. Comput. Math. Appl. 31, 91–96 (1996)
Sugiyama, K., Tagawa, S., Toda, M.: Methods for visual understanding of hierarchical system structures. IEEE Trans. Syst. Man Cybern. 11(2), 109–125 (1981). https://doi.org/10.1109/TSMC.1981.4308636
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Evans, W., Köck, K., Kobourov, S. (2024). Visualization of Bipartite Graphs in Limited Window Size. In: Fernau, H., Gaspers, S., Klasing, R. (eds) SOFSEM 2024: Theory and Practice of Computer Science. SOFSEM 2024. Lecture Notes in Computer Science, vol 14519. Springer, Cham. https://doi.org/10.1007/978-3-031-52113-3_14
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DOI: https://doi.org/10.1007/978-3-031-52113-3_14
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