### Abstract

Let G = (V, E) be a weighted undirected graph on n vertices and m edges, and let d_{G} be its shortest path metric. We present two simple deterministic algorithms for approximating all-pairs shortest paths in G. Our first algorithm runs in Õ(n^{2}) time, and for any u, v ε V reports distance no greater than 2d_{G}(u,v) + h(u,v). Here, h(u, v) is the largest edge weight on a shortest path between u and v. The previous algorithm, due to Baswana and Kavitha that achieved the same result was randomized. Our second algorithm for the all-pairs shortest path problem uses Boolean matrix multiplications and for any u, v ε V reports distance no greater than (1 + ε)d_{G}(u, v) + 2h(u, v). The currently best known algorithm for Boolean matrix multiplication yields an O(n ^{2.24+o(1)}ε^{-3}log(nε^{-1})) time bound for this algorithm. The previously best known result of Elkin with a similar multiplicative factor had a much bigger additive error term. We also consider approximating the diameter and the radius of a graph. For the problem of estimating the radius, we present an almost 3/2-approximation algorithm which runs in Õ(m√n + n^{2}) time. Aingworth, Chekuri, Indyk, and Motwani used a similar approach and obtained analogous results for the diameter approximation problem, Additionally, we show that if the graph has a small separator decomposition a 3/2-approximation of both the diameter and the radius can be obtained more efficiently.

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

Title of host publication | Algorithms and Data Structures - 10th International Workshop, WADS 2007, Proceedings |

Pages | 541-552 |

Number of pages | 12 |

State | Published - Dec 1 2007 |

Event | 10th International Workshop on Algorithms and Data Structures, WADS 2007 - Halifax, Canada Duration: Aug 15 2007 → Aug 17 2007 |

### Publication series

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

Volume | 4619 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 10th International Workshop on Algorithms and Data Structures, WADS 2007 |
---|---|

Country | Canada |

City | Halifax |

Period | 8/15/07 → 8/17/07 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Algorithms and Data Structures - 10th International Workshop, WADS 2007, Proceedings*(pp. 541-552). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4619 LNCS).

}

*Algorithms and Data Structures - 10th International Workshop, WADS 2007, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 4619 LNCS, pp. 541-552, 10th International Workshop on Algorithms and Data Structures, WADS 2007, Halifax, Canada, 8/15/07.

**Faster approximation of distances in graphs.** / Berman, Piotr; Kasiviswanathan, Shiva Prasad.

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

TY - GEN

T1 - Faster approximation of distances in graphs

AU - Berman, Piotr

AU - Kasiviswanathan, Shiva Prasad

PY - 2007/12/1

Y1 - 2007/12/1

N2 - Let G = (V, E) be a weighted undirected graph on n vertices and m edges, and let dG be its shortest path metric. We present two simple deterministic algorithms for approximating all-pairs shortest paths in G. Our first algorithm runs in Õ(n2) time, and for any u, v ε V reports distance no greater than 2dG(u,v) + h(u,v). Here, h(u, v) is the largest edge weight on a shortest path between u and v. The previous algorithm, due to Baswana and Kavitha that achieved the same result was randomized. Our second algorithm for the all-pairs shortest path problem uses Boolean matrix multiplications and for any u, v ε V reports distance no greater than (1 + ε)dG(u, v) + 2h(u, v). The currently best known algorithm for Boolean matrix multiplication yields an O(n 2.24+o(1)ε-3log(nε-1)) time bound for this algorithm. The previously best known result of Elkin with a similar multiplicative factor had a much bigger additive error term. We also consider approximating the diameter and the radius of a graph. For the problem of estimating the radius, we present an almost 3/2-approximation algorithm which runs in Õ(m√n + n2) time. Aingworth, Chekuri, Indyk, and Motwani used a similar approach and obtained analogous results for the diameter approximation problem, Additionally, we show that if the graph has a small separator decomposition a 3/2-approximation of both the diameter and the radius can be obtained more efficiently.

AB - Let G = (V, E) be a weighted undirected graph on n vertices and m edges, and let dG be its shortest path metric. We present two simple deterministic algorithms for approximating all-pairs shortest paths in G. Our first algorithm runs in Õ(n2) time, and for any u, v ε V reports distance no greater than 2dG(u,v) + h(u,v). Here, h(u, v) is the largest edge weight on a shortest path between u and v. The previous algorithm, due to Baswana and Kavitha that achieved the same result was randomized. Our second algorithm for the all-pairs shortest path problem uses Boolean matrix multiplications and for any u, v ε V reports distance no greater than (1 + ε)dG(u, v) + 2h(u, v). The currently best known algorithm for Boolean matrix multiplication yields an O(n 2.24+o(1)ε-3log(nε-1)) time bound for this algorithm. The previously best known result of Elkin with a similar multiplicative factor had a much bigger additive error term. We also consider approximating the diameter and the radius of a graph. For the problem of estimating the radius, we present an almost 3/2-approximation algorithm which runs in Õ(m√n + n2) time. Aingworth, Chekuri, Indyk, and Motwani used a similar approach and obtained analogous results for the diameter approximation problem, Additionally, we show that if the graph has a small separator decomposition a 3/2-approximation of both the diameter and the radius can be obtained more efficiently.

UR - http://www.scopus.com/inward/record.url?scp=38149066832&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38149066832&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:38149066832

SN - 3540739483

SN - 9783540739487

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

SP - 541

EP - 552

BT - Algorithms and Data Structures - 10th International Workshop, WADS 2007, Proceedings

ER -