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

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 + D0(G)), where n and n are the number of vertices and edges of the graph G, and D0(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×n grid and the running time reduces to O(n log n).

[1]  Kozo Sugiyama,et al.  Visualization of structural information: automatic drawing of compound digraphs , 1991, IEEE Trans. Syst. Man Cybern..

[2]  Robert E. Tarjan,et al.  Applications of a planar separator theorem , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[3]  János Pach,et al.  Small sets supporting fary embeddings of planar graphs , 1988, STOC '88.

[4]  Sunil Arya,et al.  Approximate range searching , 1995, SCG '95.

[5]  Arne Frick,et al.  Automatic Graph Clustering , 1996, GD.

[6]  G. W. Furnas,et al.  Generalized fisheye views , 1986, CHI '86.

[7]  Y. V. Silva-Filho.  Average case analysis of region search in balanced k-d trees , 1979 .

[8]  Sunil Arya,et al.  An optimal algorithm for approximate nearest neighbor searching fixed dimensions , 1998, JACM.

[9]  Marek Chrobak,et al.  Minimum-width grid drawings of plane graphs , 1994, Comput. Geom..

[10]  Gary L. Miller,et al.  Finding small simple cycle separators for 2-connected planar graphs. , 1984, STOC '84.

[11]  Michael T. Goodrich,et al.  Balanced aspect ratio trees: combining the advantages of k-d trees and octrees , 1999, SODA '99.

[12]  Richard J. Lipton,et al.  A method for drawing graphs , 1985, SCG '85.

[13]  Ioannis G. Tollis,et al.  Algorithms for Drawing Graphs: an Annotated Bibliography , 1988, Comput. Geom..

[14]  Walter Schnyder,et al.  Embedding planar graphs on the grid , 1990, SODA '90.

[15]  Robert F. Cohen,et al.  How to Draw a Planar Clustered Graph , 1995, COCOON.

[16]  Hanan Samet,et al.  The Design and Analysis of Spatial Data Structures , 1989 .

[17]  Alvis Brazma,et al.  A Simple Algorithm for Drawing Large Graphs on Small Screens , 1994, Graph Drawing.

[18]  Manojit Sarkar,et al.  Graphical fisheye views , 1994, CACM.

[19]  Michael R. Anderberg,et al.  Cluster Analysis for Applications , 1973 .

[20]  S. Rao Kosaraju,et al.  A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields , 1995, JACM.

[21]  K. Wagner.  Bemerkungen zum Vierfarbenproblem. , 1936 .

[22]  Peter Eades,et al.  Multilevel Visualization of Clustered Graphs , 1996, GD.

[23]  John A. Hartigan,et al.  Clustering Algorithms , 1975 .

[24]  Fan Chung,et al.  On the Decomposition of Graphs , 1981 .

[25]  Jon Louis Bentley,et al.  An Algorithm for Finding Best Matches in Logarithmic Expected Time , 1977, TOMS.

[26]  Robert F. Cohen,et al.  Planarity for Clustered Graphs , 1995, ESA.