### Abstract

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 × n grid and the running time reduces to O(n log n).

Original language | English (US) |
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Title of host publication | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Publisher | Springer Verlag |

Pages | 111-124 |

Number of pages | 14 |

Volume | 1547 |

ISBN (Print) | 3540654739, 9783540654735 |

DOIs | |

State | Published - 1999 |

Externally published | Yes |

Event | 6th International Symposium on Graph Drawing, GD 1998 - Montreal, Canada Duration: Aug 13 1998 → Aug 15 1998 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
---|---|

Volume | 1547 |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 6th International Symposium on Graph Drawing, GD 1998 |
---|---|

Country | Canada |

City | Montreal |

Period | 8/13/98 → 8/15/98 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

### Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*(Vol. 1547, pp. 111-124). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1547). Springer Verlag. https://doi.org/10.1007/3-540-37623-2_9

**Balanced aspect ratio trees and their use for drawing very large graphs.** / Duncan, Christian A.; Goodrich, Michael T.; Kobourov, Stephen G.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics).*vol. 1547, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 1547, Springer Verlag, pp. 111-124, 6th International Symposium on Graph Drawing, GD 1998, Montreal, Canada, 8/13/98. https://doi.org/10.1007/3-540-37623-2_9

}

TY - GEN

T1 - Balanced aspect ratio trees and their use for drawing very large graphs

AU - Duncan, Christian A.

AU - Goodrich, Michael T.

AU - Kobourov, Stephen G

PY - 1999

Y1 - 1999

N2 - 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 m 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).

AB - 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 m 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).

UR - http://www.scopus.com/inward/record.url?scp=84957868512&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84957868512&partnerID=8YFLogxK

U2 - 10.1007/3-540-37623-2_9

DO - 10.1007/3-540-37623-2_9

M3 - Conference contribution

AN - SCOPUS:84957868512

SN - 3540654739

SN - 9783540654735

VL - 1547

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 111

EP - 124

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

PB - Springer Verlag

ER -