Abstract
This paper analyzes the exact and asymptotic worst-case complexity of the simplification phase of Sugiyama's algorithm [12] for drawing arbitrary directed graphs.
The complexity of this phase is determined by the number of hidden nodes inserted. The best previously known upper bound for this number is O(max{¦V¦3,¦E¦2}). This paper establishes a relation between both partial results and gives upper bounds for many classes of graphs. This is achieved by constructing a worst-case example for every legal configuration C=(h, n, m) of the input hierarchy for the simplification phase. These results provide further insight into the worst-case runtime and space complexity of Sugiyama's algorithm. Possible applications include their use as feasibility criteria, based on simply derived quantitative information on the graph.
This work was performed while the author was employed at Universität Karlsruhe, Institut für Programmstrukturen und Datenorganisation.
Chapter PDF
References
G. di Battista, P. Eades, R. Tamassia, and I. G. Tollis. Algorithms for drawing graphs: an annotated bibliography. Report, June 1993.
P. Eades and K. Sugiyama. How to draw a directed graph. Journal of Information Processing, 14(4):424–437, 1990.
M. Fröhlich and M. Werner. Demonstration of the interactive graph-visualization system davinci. In Tamassia and Tollis [13].
E. R. Gansner, E. Koutsofios, S. C. North, and K.-P. Vo. A technique for drawing directed graphs. IEEE Transactions on Software Engineering, 19(3):214–230, March 1993.
F. Harary. Graph Theory. Series in Mathematics. Addison Wesley Publishing Company, 1969.
M. Himsolt. Graphed: A graphical platform for the implementation of graph algorithms. In Tamassia and Tollis [13].
I. Lemke. Entwicklung und Implementierung eines Visualisierungswerkzeuges für anwendungen im Übersetzerbau. Diplomarbeit, Universität des Saarlandes, FB 14 Informatik, 1994.
B. Madden, P. Madden, S. Powers, and M. Himsolt. Portable graph layout and editing (system demonstration). In Franz Brandenburg, editor, Proceedings of Graph Drawing '95, volume 1027 of Lecture Notes in Computer Science, pages 385–395. Springer Verlag, 1996.
F. Newbery-Paulisch and W. F. Tichy. Edge: An extendible graph editor. Software — Practic and Experience, 20(S1):S1/63–S1/88, June 1990.
G. Sander. Graph layout through the VCG tool. In Tamassia and Tollis [13], pages 194–205.
G. Sander. Visualisierungstechniken fuer den Compilerbau. PhD thesis, Univ. des Saarlandes, FB 14 Informatik, Saarbrücken, 1996.
K. Sugiyama, S. Tagawa, and M. Toda. Methods for visual understanding of hierarchical system structures. IEEE Transactions on Systems, Man and Cybernetics, SMC-11(2):109–125, February 1981.
R. Tamassia and I. Tollis, editors. Proceedings of Graph Drawing '94, volume 894 of Lecture Notes in Computer Science. DIMACS Workshop on Graph Drawing, Springer Verlag, 1995.
Waterloo Maple Software. Maple V Release 3, 1994.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Frick, A. (1997). Upper bounds on the number of hidden nodes in Sugiyama's algorithm. In: North, S. (eds) Graph Drawing. GD 1996. Lecture Notes in Computer Science, vol 1190. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-62495-3_46
Download citation
DOI: https://doi.org/10.1007/3-540-62495-3_46
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-62495-0
Online ISBN: 978-3-540-68048-2
eBook Packages: Springer Book Archive