Proximity Drawings: Three Dimensions Are Better than Two
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We consider weak Gabriel drawings of unbounded degree trees in the three-dimensional space. We assume a minimum distance between any two vertices. Under the same assumption, there exists an exponential area lower bound for general graphs. Moreover, all previously known algorithms to construct (weak) proximity drawings of trees, generally produce exponential area layouts, even when we restrict ourselves to binary trees. In this paper we describe a linear-time polynomial-volume algorithm that constructs a strictly-upward weak Gabriel drawing of any rooted tree with O(log n)-bit requirement. As a special case we describe a Gabriel drawing algorithm for binary trees which produces integer coordinates and n 3-area representations. Finally, we show that an infinite class of graphs requiring exponential area, admits linear-volume Gabriel drawings. The latter result can also be extended to β-drawings, for any 1 < β < 2, and relative neighborhood drawings.
- 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. CrossRef
- G. Di Battista, W. Lenhart, and G. Liotta. Proximity drawability: a survey. In Proc. Graph Drawing’ 94, Lecture Notes in Computer Science, pages 328–339. Springer Verlag, 1994.
- G. DiBattista, G. Liotta, and S.H. Whitesides. The strenght of weak proximity. In F. J. Brandenburg, editor, Graph Drawing (Proc. GD’ 95), volume 1027 of itLecture Notes in Computer Science, pages 178–189. Springer-Verlag, 1996. CrossRef
- P. Bose, G. Di Battista, W. Lenhart, and G. Liotta. Proximity constraints and representable trees. In Proc. Graph Drawing rs94, LNCS, pages 340–351. Springer-Verlag, 1994.
- P. Bose, W. Lenhart, and G. Liotta. Characterizing proximity trees. Algorithmica, 16:83–110, 1996. CrossRef
- M. Brown and M. Najork. Algorithm animation using 3d interactive graphics. In Proc. ACM Symp. on User Interface Software and Technology, pages 93–100, 1993.
- T. Calamoneri and A. Sterbini. Drawing 2-, 3-and 4-colorable graphs in o( n 2) volume. In Stephen North, editor, Graph Drawing (Proc. GD’ 96 ), volume 1190 of Lecture Notes in Computer Science, pages 53–62. Springer-Verlag, 1997.
- 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 rs94, volume 894 of Lecture Notes in Computer Science, pages 1–11. Springer-Verlag, 1995.
- P. Eades and S. Whitesides. The realization problem for euclidean minimum spanning tree is NP-hard. In Proc. ACM Symp. on Computational Geometry, pages 49–56, 1994.
- H. ElGindy, G. Liotta, A. Lubiw, H. Meijer, and S. H. Whitesides. Recognizing rectangle of influence drawable graphs. In Proc. Graph Drawing rs94, number LNCS, pages 352–363. Springer-Verlag, 1994.
- K. R. Gabriel and R. R. Sokal. A new statistical approach to geographic variation analysis. Systematic Zoology, 18:259–278, 1969. CrossRef
- A. Garg and R. Tamassia. GIOTTO3D: A system for visualizing hierarchical structures in 3d. In Stephen North, editor, Graph Drawing (Proc. GD)’ 96, volume 1190 of Lecture Notes in Computer Science, pages 193–200, 1997.
- D.G. Kirkpatrick and J.D. Radke. A framework for computational morphology. In G.T. Toussaint, editor, Computational Geometry, pages 217–248, Amsterdam, Netherlands, 1985. North-Holland.
- G. Liotta and G. Di Battista. Computing proximity drawings of trees in the 3-dimensional space. In Proc. 4th Workshop Algorithms Data Struct, volume 955 of Lecture Notes in Computer Science, pages 239–250. Springer-Verlag, 1995.
- G. Liotta, R. Tamassia, J. G. Tollis, and P. Vocca. Area requirement of gabriel drawings. In Giancarlo Bongiovanni, Daniel Pierre Bovet, and Giuseppe Di Battista, editors, Proc. CIACrs97, volume 1203 of Lecture Notes in Computer Science, pages 135–146. Spriger-Verlag, 1997.
- A. Lubiw and N. Sleumer. All maximal outerplanar graphs are relative neighborhood graphs. In Proc. CCCGrs93, pages 198–203, 1993.
- J. MacKinley, G. Robertson, and S. Card. Cone trees: Animated 3d visualizations of hierarchical information. In Proc. of SIGCHI Conf. on Human Factors in Computing, pages 189–194, 1991.
- D. W. Matula and R. R. Sokal. Properties of gabriel graphs relevant to geographic variation research and clustering of points in the plane. Geogr. Anal., 12:205–222, 1980. CrossRef
- M. Patrignani and F. Vargiu. 3DCube: A tool for three dimensional graph drawing. In Giseppe Di Battista, editor, Graph Drawing (Proc. GD rs97), volume 1353 of Lecture Notes in Computer Science, pages 284–290. Springer-Verlag, 1997. CrossRef
- J.D. Radke. On the shape of a set of points. In G.T. Toussaint, editor, Computational Morphology, pages 105–136, Amsterdam, The Netherlands, 1988. North-Holland.
- S. Reiss. An engine for the 3D visualization of program information. J. Visual Languages and Computing, 6(3), 1995.
- G.T. Toussaint. The relative neighborhood graph of a finite planar set. Pattern recognition, 12:261–268, 1980. CrossRef
- Proximity Drawings: Three Dimensions Are Better than Two
- Book Title
- Graph Drawing
- Book Subtitle
- 6th International Symposium, GD’ 98 Montréal, Canada, August 13–15, 1998 Proceedings
- pp 275-287
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- Series Title
- Lecture Notes in Computer Science
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- Springer Berlin Heidelberg
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- Springer-Verlag Berlin Heidelberg
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- Editor Affiliations
- 4. School of Computer Science, McGill University
- Author Affiliations
- 5. Dipartimento di Scienze dell’Informazione, Università di Roma “La Sapienza’, Via Salaria 113, I-00198, Rome, Italy
- 6. Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica, I-00133, Rome, Italy
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