Balanced Aspect Ratio Trees and Their Use for Drawing Very Large Graphs

Duncan, Christian A. and Goodrich, Michael T. and Kobourov, Stephen G. (1998) Balanced Aspect Ratio Trees and Their Use for Drawing Very Large Graphs. In: Graph Drawing 6th International Symposium, GD’ 98, August 13-15, 1998, Montréal, Canada , pp. 111-124 (Official URL:

Full text not available from this repository.


We describe a new approach for cluster-based drawing of very large graphs, which obtains clusters by using binary space partition (BSP) trees. We also introduce a novel BSP-type decomposition, called the balanced aspect ratio (BAR) tree, which guarantees that the cells produced are convex and have bounded aspect ratios. In addition, the tree depth is O(log n), and its construction takes O(n log n) time, where n is the number of points. We show that the BAR tree can be used to recursively divide a graph into subgraphs of roughly equal size, such that the drawing of each subgraph has a balanced aspect ratio. As a result, we obtain a representation of a graph as a collection of O(log n) layers, where each succeeding layer represents the graph in an increasing level of detail. The overall running time of the algorithm is O(n log n+m+D_0(G)), where n and m are the number of vertices and edges of the graph G, and D_0(G) is the time it takes to obtain an initial embedding of G. In particular, if the graph is planar each layer is a graph drawn with straight lines and without crossings on the n \times n grid and the running time reduces to O(n log n).

Item Type:Conference Paper
Additional Information:10.1007/3-540-37623-2_9
Classifications:G Algorithms and Complexity > G.700 Layering
G Algorithms and Complexity > G.999 Others
G Algorithms and Complexity > G.070 Area / Edge Length
G Algorithms and Complexity > G.350 Clusters
ID Code:264

Repository Staff Only: item control page


S. Arya, D.M. Mount, N.S. Netananyahu, R. Silverman, and A. Wu. An optimal algorithm for approximate nearest neighbor searching. In Proc. 5th ACM-SIAM Sympos. Discrete Algorithms, pages 573-582, 1994.

S. Arya and D.M. Mount. Approximate range searching. In Proc. 11th Annu. ACM Sympos. Comput. Geom., pages 172-181, 1995.

P. Callahan and S.R. Kosaraju. A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields. J. ACM, 42:67-90, 1995.

H. de Fraysseix, J. Pach, and R. Pollack. Small sets supporting Fary embeddings of planar graphs. In Proc. 20th Annu. ACM Sympos. Theory Comput., pages 426-433, 1988.

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.

Peter Eades and Qing-Wen Feng. Multilevel Visualization of Clustered Graphs. LNCS, 1190:101-112, 1997.

Peter Eades, Qing-Wen Feng, and Xuemin Lin. Straight-line drawing algorithms for hierarchical graphs and clustered graphs. LNCS, 1190:113-128, 1997.

I. Fary. On straight lines representation of planar graphs. Acta Sci. Math. Szeged., 11:229-233, 1948.

Qing-Wen Feng, Robert F. Cohen, and Peter Eades. How to draw a planar clustered graph. In COCOON '95, volume 959 of LNCS, pages 21-31. Springer-Verlag, 1995.

Qing-Wen Feng, Robert F. Cohen, and Peter Eades. Planarity for clustered graphs. In ESA '95, volume 979 of LNCS, pages 213-226. Springer-Verlag, 1995.

J.H. Friedman, J.L. Bentley, and R.A. Finkel. An algorithm for finding best matches in logeritmic expected time. ACM Trans. Math. Softw., 3:209-226, 1977.

12. G.W. Furnas. Generalized fisheye views. In Proc. of ACM CHI `86 Conference on Human Factors in Computing Systems, Visualizing Complex Information Spaces, pages 16-23, 1986.

K. Kaugars, J. Reinfelds, and A. Brazma. A simple algorithm for drawing large graphs on small screens. LNCS, 894:278-281, Springer-Verlag, 1995.

R.J. Lipton, S.C. North, and J.S. Sandberg. A method for drawing graphs. In 'proc. 1st Annu ACM Sympos. Comput. Geom., pages 153-160, 1985.

R.J. Lipton and R.E. Tarjan. Applications of a planar separator theorem. SIAM j. Comput., 9:615-627, 1980.

G.L. Miller. Finding small simple cycle separators for 2-connected planar graphs. Raport 85-336, Dept. Comput. Sci., Univ. Southern California, Los Angeles, CA, 1985.

F. Newbery. Edge concentration: A method for clustering directed graphs. In Proc. of th 2nd International Workshop on Software Configuration Management, pages 76-85, Princeton, New Jersey, October 1989.

Stephen C. North. Drawing ranked digraphs with recursive clusters. In Proc. ALCOM Workshop on Graph Drawing '93, September 1993.

Sablowski and Frick. Automatic graph clustering. In GDrawing (GD '96), 1996.

H. Samet. The Design and Analysis of Spatial Data Structures. Addison-Wesley, Reading, MA, 1990.

M. Sarkar, and M. Brown. Graphical Fisheye Views. Commun. ACM, 37(12):73-84, 1994.

W. Schnyder. Embedding planar graphs on the grid. In Proc. 1st ACM-SIAM Symp. Discrete Algorithms, pages 138-148, 1990.

S.K. Stein. Convex maps. Proc. Amer. Math. Soc., 2:464-466, 1951.

K. Sugiyama and K. Misue. Visualization of structural information: Automatic drawing of compound digraphs. IEEE Transactions on Systems, Man and Cybernetics, 21(4):876-892, 1991.

W.T. Tutte. How to draw a graph. Proceedings of the London Mathematical Society, 3(13):743-768, 1963.

K. Wagner. Bemerkungen zum vierfarben problem. Jber. Deutsch. Math.-Verein,46:26-32, 1936.