CHI '86 Proceedings of the SIGCHI Conference on Human Factors in Computing Systems
Small sets supporting fary embeddings of planar graphs
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Edge concentration: a method for clustering directed graphs
SCM '89 Proceedings of the 2nd International Workshop on Software configuration management
The design and analysis of spatial data structures
The design and analysis of spatial data structures
Algorithms for drawing graphs: an annotated bibliography
Computational Geometry: Theory and Applications
Communications of the ACM
Proceedings of the eleventh annual symposium on Computational geometry
An optimal algorithm for approximate nearest neighbor searching
SODA '94 Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms
Embedding planar graphs on the grid
SODA '90 Proceedings of the first annual ACM-SIAM symposium on Discrete algorithms
SCG '85 Proceedings of the first annual symposium on Computational geometry
An Algorithm for Finding Best Matches in Logarithmic Expected Time
ACM Transactions on Mathematical Software (TOMS)
How to Draw a Planar Clustered Graph
COCOON '95 Proceedings of the First Annual International Conference on Computing and Combinatorics
GD '96 Proceedings of the Symposium on Graph Drawing
Straight-Line Drawing Algorithms for Hierarchical Graphs and Clustered Graphs
GD '96 Proceedings of the Symposium on Graph Drawing
A Simple Algorithm for Drawing Large Graphs on Small Screens
GD '94 Proceedings of the DIMACS International Workshop on Graph Drawing
Planarity for Clustered Graphs
ESA '95 Proceedings of the Third Annual European Symposium on Algorithms
Multilevel Visualization of Clustered Graphs
GD '96 Proceedings of the Symposium on Graph Drawing
Balanced aspect ratio trees: combining the advantages of k-d trees and octrees
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
Large networks present visualization challenges
ACM SIGGRAPH Computer Graphics
Graph Visualization and Navigation in Information Visualization: A Survey
IEEE Transactions on Visualization and Computer Graphics
MGV: A System for Visualizing Massive Multidigraphs
IEEE Transactions on Visualization and Computer Graphics
Planarity-preserving clustering and embedding for large planar graphs
Computational Geometry: Theory and Applications - Fourth CGC workshop on computional geometry
Hierarchical Clustering of Trees: Algorithms and Experiments
ALENEX '01 Revised Papers from the Third International Workshop on Algorithm Engineering and Experimentation
A Multi-dimensional Approach to Force-Directed Layouts of Large Graphs
GD '00 Proceedings of the 8th International Symposium on Graph Drawing
K-D Trees Are Better when Cut on the Longest Side
ESA '00 Proceedings of the 8th Annual European Symposium on Algorithms
Range Searching Over Tree Cross Products
ESA '00 Proceedings of the 8th Annual European Symposium on Algorithms
Visualizing Massive Multi-Digraphs
INFOVIS '00 Proceedings of the IEEE Symposium on Information Vizualization 2000
Effective Graph Visualization Via Node Grouping
INFOVIS '01 Proceedings of the IEEE Symposium on Information Visualization 2001 (INFOVIS'01)
Proceedings of the Working Conference on Advanced Visual Interfaces
Balanced aspect ratio trees revisited
WADS'05 Proceedings of the 9th international conference on Algorithms and Data Structures
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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脳n grid and the running time reduces to O(n log n).