### Abstract

We describe a new approach for cluster-based drawing of 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 embedded in the plane 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 the plane. 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) |
---|---|

Pages (from-to) | 19-46 |

Number of pages | 28 |

Journal | Journal of Graph Algorithms and Applications |

Volume | 4 |

Issue number | 3 |

State | Published - 2000 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science
- Computer Science(all)
- Computer Science Applications

### Cite this

*Journal of Graph Algorithms and Applications*,

*4*(3), 19-46.

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

Research output: Contribution to journal › Article

*Journal of Graph Algorithms and Applications*, vol. 4, no. 3, pp. 19-46.

}

TY - JOUR

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

AU - Duncan, Christian A.

AU - Goodrich, Michael T.

AU - Kobourov, Stephen G

PY - 2000

Y1 - 2000

N2 - We describe a new approach for cluster-based drawing of 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 embedded in the plane 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 the plane. 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 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 embedded in the plane 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 the plane. 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=4043049304&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4043049304&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:4043049304

VL - 4

SP - 19

EP - 46

JO - Journal of Graph Algorithms and Applications

JF - Journal of Graph Algorithms and Applications

SN - 1526-1719

IS - 3

ER -