### Abstract

Optimizing a stress model is a natural technique for drawing graphs: one seeks an embedding into R^{d} which best preserves the induced graph metric. Current approaches to solving the stress model for a graph with |ν| nodes and |ε| edges require the full all-pairs shortest paths (APSP) matrix, which takes O(|ν|2 log |ε|+|ν||ε|) time and O(|ν|^{2}) space. We propose a novel algorithm based on a low-rank approximation to the required matrices. The crux of our technique is an observation that it is possible to approximate the full APSP matrix, even when only a small subset of its entries are known. Our algorithm takes time O(k|ν|+|ν| log|ν|+|ε|) per iteration with a preprocessing time of O(k^{3} +k(|ε|+|ν| log|ν|)+k2|ν|) and memory usage of O(k|ν|), where a user-defined parameter k trades off quality of approximation with running time and space. We give experimental results which show, to the best of our knowledge, the largest (albeit approximate) full stress model based layouts to date. Computer Graphics Forum

Original language | English (US) |
---|---|

Pages (from-to) | 975-984 |

Number of pages | 10 |

Journal | Computer Graphics Forum |

Volume | 31 |

Issue number | 3 PART1 |

State | Published - 2012 |

Externally published | Yes |

### Fingerprint

### ASJC Scopus subject areas

- Computer Networks and Communications

### Cite this

*Computer Graphics Forum*,

*31*(3 PART1), 975-984.

**Drawing large graphs by low-rank stress majorization.** / Khoury, Marc; Hu, Yifan; Krishnan, Shankar; Scheidegger, Carlos Eduardo.

Research output: Contribution to journal › Article

*Computer Graphics Forum*, vol. 31, no. 3 PART1, pp. 975-984.

}

TY - JOUR

T1 - Drawing large graphs by low-rank stress majorization

AU - Khoury, Marc

AU - Hu, Yifan

AU - Krishnan, Shankar

AU - Scheidegger, Carlos Eduardo

PY - 2012

Y1 - 2012

N2 - Optimizing a stress model is a natural technique for drawing graphs: one seeks an embedding into Rd which best preserves the induced graph metric. Current approaches to solving the stress model for a graph with |ν| nodes and |ε| edges require the full all-pairs shortest paths (APSP) matrix, which takes O(|ν|2 log |ε|+|ν||ε|) time and O(|ν|2) space. We propose a novel algorithm based on a low-rank approximation to the required matrices. The crux of our technique is an observation that it is possible to approximate the full APSP matrix, even when only a small subset of its entries are known. Our algorithm takes time O(k|ν|+|ν| log|ν|+|ε|) per iteration with a preprocessing time of O(k3 +k(|ε|+|ν| log|ν|)+k2|ν|) and memory usage of O(k|ν|), where a user-defined parameter k trades off quality of approximation with running time and space. We give experimental results which show, to the best of our knowledge, the largest (albeit approximate) full stress model based layouts to date. Computer Graphics Forum

AB - Optimizing a stress model is a natural technique for drawing graphs: one seeks an embedding into Rd which best preserves the induced graph metric. Current approaches to solving the stress model for a graph with |ν| nodes and |ε| edges require the full all-pairs shortest paths (APSP) matrix, which takes O(|ν|2 log |ε|+|ν||ε|) time and O(|ν|2) space. We propose a novel algorithm based on a low-rank approximation to the required matrices. The crux of our technique is an observation that it is possible to approximate the full APSP matrix, even when only a small subset of its entries are known. Our algorithm takes time O(k|ν|+|ν| log|ν|+|ε|) per iteration with a preprocessing time of O(k3 +k(|ε|+|ν| log|ν|)+k2|ν|) and memory usage of O(k|ν|), where a user-defined parameter k trades off quality of approximation with running time and space. We give experimental results which show, to the best of our knowledge, the largest (albeit approximate) full stress model based layouts to date. Computer Graphics Forum

UR - http://www.scopus.com/inward/record.url?scp=84879713276&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84879713276&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:84879713276

VL - 31

SP - 975

EP - 984

JO - Computer Graphics Forum

JF - Computer Graphics Forum

SN - 0167-7055

IS - 3 PART1

ER -