Abstract
In this paper, we investigate the volume, aspect ratio, angular resolution, edge-separation, and bit-requirement of crossing-free straight-line 3D drawings. We assume the vertex resolution rule, which requires minimum unit distance between any two vertices. Our main result shows that an N-vertex graph colorable with O(1) colors admits a crossing-free straight-line 3D drawing with O(N√N) volume, O(1) aspect ratio,gW(l/N O(1)) angular resolution, Ω (1/N O(1)) edge-separation, and O(log N) bit-requirement, which can be constructed in O(N) time.
Research supported in part the National Science Foundation under grant CCR-9423847, by the U.S. Army Research Office under grant DAAH04-96-1-0013, and by a gift from Tom Sawyer Software. Research performed in part while Paola Vocca was visiting Brown University, and while Roberto Tamassia and Paola Vocca were visiting the Bellairs Research Institute of McGill University.
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H. Alt, M. Godau, and S. Whitesides. Universal 3-dimensional visibility representations for graphs. In F. J. Brandenburg, editor, Graph Drawing (Proc. GD '95), volume 1027 of Lecture Notes in Computer Science, pages 8–19. Springer-Verlag, 1996.
I. Bruß and A. Frick. Fast interactive 3-d graph visualization. In F. Brandenburg, editor, Graph Drawing (Proc. GD '95), volume 1027 of LNCS, pages 99–110, 1996.
T. Calamoneri and A. Sterbini. Drawing 2-, 3-and 4-colorable graphs in o(n 2) volume. Technical report, Dept. of Comp. Sc., Univ. Rome “La Sapienza”, 1996.
M. Chrobak, M. T. Goodrich, and R. Tamassia. Convex drawings of graphs in two and three dimensions. In Proc. 12th Annu. ACM Sympos. Comput. Geom., pages 319–328, 1996.
R. F. Cohen, P. Eades, T. Lin, and F. Ruskey. Three-dimensional graph drawing. In R. Tamassia and I. G. Tollis, editors, Graph Drawing (Proc. GD '94), volume 894 of Lecture Notes in Computer Science, pages 1–11. Springer-Verlag, 1995.
I. Cruz and J. Twarog. 3d graph drawing with simulated annealing. In F. Brendenburg, editor, Graph Drawing (Proc. GD '95), volume 1027 of Lecture Notes in Computer Science, pages 162–165. Springer-Verlag, 1996.
G. Das and M. T. Goodrich. On the complexity of approximating and illuminating three-dimensional convex polyhedra. In Proc. 4th Workshop Algorithms Data Struct., volume 955 of Lecture Notes in Computer Science, pages 74–85. Springer-Verlag, 1995.
G. Di Battista, P. Eades, R. Tamassia, and I. G. Tollis. Algorithms for drawing graphs: an annotated bibliography. Comput. Geom. Theory Appl, 4:235–282, 1994.
P. Eades and P. Garvan. Drawing stressed planar graphs in three dimensions. In F. J. Brandenburg, editor, Graph Drawing (Proc. GD '95), volume 1027 of Lecture Notes in Computer Science. Springer-Verlag, 1996.
P. Eades, C. Stirk, and S. Whitesides. The techniques of Komolgorov and Bardzin for three dimensional orthogonal graph drawings. Manuscript, Dept. of Computer Sci., Univ. of Newcastle, 1995.
M. Formann, T. Hagerup, J. Haralambides, M. Kaufmann, F. T. Leighton, A. Simvonis, E. Welzl, and G. Woeginger. Drawing graphs in the plane with high resolution. SIAM J. Comput, 22:1035–1052, 1993.
A. Garg and R. Tamassia. Area-optimal upward tree drawings. Int. Journal of Computational Geometry: Theory and Applns. to appear.
S. M. Hashemi and I. Rival. Upward drawings to fit surfaces. In Order, Algorithms, and Applications (Proc. ORDAL '94), volume 831 of Lecture Notes in Computer Science, pages 53–58. Springer-Verlag, 1994.
D. Jablonowsky and V. A. Guarna. GMB: A tool for manipulating and animating graph data structures. Softw. — Pract. Exp., 19(3):283–301, 1989.
T. Jéron and C. Jard. 3D layout of reachability graphs of communicating processes. In R. Tamassia and I. G. Tollis, editors, Graph Drawing (Proc. GD '94), volume 894 of Lecture Notes in Computer Science, pages 25–32. Springer-Verlag, 1995.
F. T. Leighton and A. Rosenberg. 3d circuit layouts. SIAM J. Comput., 15:793–813, 1986.
G. Liotta and G. Di Battista. Computing proximity drawings of trees in the 3-dimemsional space. In Proc. 4th Workshop Algorithms Data Struct., volume 955 of Lecture Notes in Computer Science, pages 239–250. Springer-Verlag, 1995.
B. Monien, F. Ramme, and H. Salmen. A parallel simulated annealing algorithm for generating 3d layouts of undirected graphs. In F. Brandenburg, editor, Graph Drawing (Proc. GD '95), volume 1027 of LNCS, pages 396–408, 1996.
S. P. Reiss. An engine for the 3D visualization of program information. J. Visual Languages and Computing, 6(3), 1995. (special issue on Graph Visualization, edited by I. F. Cruz and P. Eades).
G. G. Robertson, J. D. Mackinlay, and S. K. Card. Cone trees: Animated 3d visualizations of hierarchical information. In Proc. CHI, pages 189–193, 1991.
A. L. Rosenberg. Three-dimensional VLSI: a case study. J. ACM, 30(3):397–416, 1983.
E. Steinitz and H. Rademacher. Vorlesungen über die Theorie der Polyeder. Julius Springer, Berlin, Germany, 1934.
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Garg, A., Tamassia, R., Vocca, P. (1996). Drawing with colors. In: Diaz, J., Serna, M. (eds) Algorithms — ESA '96. ESA 1996. Lecture Notes in Computer Science, vol 1136. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61680-2_43
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DOI: https://doi.org/10.1007/3-540-61680-2_43
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