Abstract
We show that a transitively reduced digraph has a confluent upward drawing if and only if its reachability relation has order dimension at most two. In this case, we construct a confluent upward drawing with O(n 2) features, in an O(n) ×O(n) grid in O(n 2) time. For the digraphs representing series-parallel partial orders we show how to construct a drawing with O(n) features in an O(n) ×O(n) grid in O(n) time from a series-parallel decomposition of the partial order. Our drawings are optimal in the number of confluent junctions they use.
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Aeschlimann, A., Schmid, J.: Drawing orders using less ink. Order 9(1), 5–13 (1992)
Amer, P., Chassot, C., Connolly, T., Diaz, M., Conrad, P.: Partial-order transport service for multimedia and other applications. IEEE/ACM Transactions on Networking 2(5), 440–456 (1994)
Baker, K.A., Fishburn, P.C., Roberts, F.S.: Partial orders of dimension 2. Networks 2(1), 11–28 (1972)
Bertolazzi, P., Di Battista, G., Mannino, C., Tamassia, R.: Optimal upward planarity testing of single-source digraphs. SIAM J. Comput. 27(1), 132–169 (1998)
Birkhoff, G.: Lattice Theory. American Mathematical Society, Providence (1967)
Di Battista, G., Tamassia, R.: Algorithms for plane representations of acyclic digraphs. Theoret. Comput. Sci. 61(2-3), 175–198 (1988)
Dickerson, M.T., Eppstein, D., Goodrich, M.T., Meng, J.Y.: Confluent drawings: visualizing non-planar diagrams in a planar way. J. Graph Algorithms Appl. 9(1), 3–52 (2005), http://jgaa.info/accepted/2005/Dickerson+2005.9.1.pdf
Eppstein, D., Goodrich, M.T., Meng, J.Y.: Delta-Confluent Drawings. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 165–176. Springer, Heidelberg (2006)
Eppstein, D., Goodrich, M.T., Meng, J.Y.: Confluent layered drawings. Algorithmica 47(4), 439–452 (2007)
Gansner, E., Hu, Y., North, S., Scheidegger, C.: Multilevel agglomerative edge bundling for visualizing large graphs. In: IEEE Pacific Visualization Symposium (PacificVis), pp. 187–194 (2011)
Ganter, B., Kuznetsov, S.O.: Stepwise Construction of the Dedekind-MacNeille Completion. In: Mugnier, M.-L., Chein, M. (eds.) ICCS 1998. LNCS (LNAI), vol. 1453, pp. 295–302. Springer, Heidelberg (1998)
Garg, A., Tamassia, R.: On the computational complexity of upward and rectilinear planarity testing. SIAM J. Comput. 31(2), 601–625 (2002)
Güting, R.H., Nurmi, O., Ottmann, T.: Fast algorithms for direct enclosures and direct dominances. J. Algorithms 10(2), 170–186 (1989)
de la Higuera, C., Nourine, L.: Drawing and encoding two-dimensional posets. Theoret. Comput. Sci. 175(2), 293–308 (1997)
Hirsch, M., Meijer, H., Rappaport, D.: Biclique Edge Cover Graphs and Confluent Drawings. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS, vol. 4372, pp. 405–416. Springer, Heidelberg (2007)
Hui, P., Pelsmajer, M.J., Schaefer, M., Štefankovič, D.: Train tracks and confluent drawings. Algorithmica 47(4), 465–479 (2007)
Hutton, M.D., Lubiw, A.: Upward planar drawing of single source acyclic digraphs. SIAM J. Comput. 25(2), 291–311 (1996)
Jourdan, G.V., Rival, I., Zaguia, N.: Upward Drawing on the Plane Grid Using Less Ink. In: Tamassia, R., Tollis, I.G. (eds.) GD 1994. LNCS, vol. 894, pp. 318–327. Springer, Heidelberg (1995)
Kelly, D., Rival, I.: Planar lattices. Canad. J. Math. 27(3), 636–665 (1975)
Lodaya, K., Weil, P.: Series-Parallel Posets: Algebra, Automata and Languages. In: Meinel, C., Morvan, M. (eds.) STACS 1998. LNCS, vol. 1373, pp. 555–565. Springer, Heidelberg (1998)
Ma, T.H., Spinrad, J.: Transitive closure for restricted classes of partial orders. Order 8(2), 175–183 (1991)
MacNeille, H.M.: Partially ordered sets. Trans. Amer. Math. Soc. 42(3), 416–460 (1937)
Mannila, H., Meek, C.: Global partial orders from sequential data. In: Proceedings of the sixth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD 2000, pp. 161–168. ACM, New York (2000)
Möhring, R.H.: Computationally tractable classes of ordered sets. In: Rival, I. (ed.) Algorithms and Order, pp. 105–193. Kluwer Academic Publishers (1989)
Möhring, R.H., Schäffter, M.W.: Scheduling series-parallel orders subject to 0/1-communication delays. Parallel Comput. 25(1), 23–40 (1999)
Nourine, L., Raynaud, O.: A fast algorithm for building lattices. Inform. Process. Lett. 71(5-6), 199–204 (1999)
Novák, V.: Über eine Eigenschaft der Dedekind-MacNeilleschen Hülle. Math. Ann. 179, 337–342 (1969)
Platt, C.R.: Planar lattices and planar graphs. J. Combinatorial Theory, Ser. B 21(1), 30–39 (1976)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Eppstein, D., Simons, J.A. (2012). Confluent Hasse Diagrams. In: van Kreveld, M., Speckmann, B. (eds) Graph Drawing. GD 2011. Lecture Notes in Computer Science, vol 7034. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25878-7_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-25878-7_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-25877-0
Online ISBN: 978-3-642-25878-7
eBook Packages: Computer ScienceComputer Science (R0)