### Abstract

We give an -approximation algorithm for the problem of finding the sparsest spanner of a given directed graph G on n vertices. A spanner of a graph is a sparse subgraph that approximately preserves distances in the original graph. More precisely, given a graph G∈=∈(V,E) with nonnegative edge lengths d: E → ℝ^{≥0} and a stretch k ≥ 1, a subgraph H = (V,E _{H} ) is a k-spanner of G if for every edge (u,v)ε E, the graph H contains a path from u to v of length at most k •d(u,v). The previous best approximation ratio was , due to Dinitz and Krauthgamer (STOC '11). We also present an improved algorithm for the important special case of directed 3-spanners with unit edge lengths. The approximation ratio of our algorithm is which almost matches the lower bound shown by Dinitz and Krauthgamer for the integrality gap of a natural linear programming relaxation. The best previously known algorithms for this problem, due to Berman, Raskhodnikova and Ruan (FSTTCS '10) and Dinitz and Krauthgamer, had approximation ratio .

Original language | English (US) |
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Title of host publication | Automata, Languages and Programming - 38th International Colloquium, ICALP 2011, Proceedings |

Pages | 1-12 |

Number of pages | 12 |

Edition | PART 1 |

DOIs | |

State | Published - Jul 11 2011 |

Event | 38th International Colloquium on Automata, Languages and Programming, ICALP 2011 - Zurich, Switzerland Duration: Jul 4 2011 → Jul 8 2011 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Number | PART 1 |

Volume | 6755 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 38th International Colloquium on Automata, Languages and Programming, ICALP 2011 |
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Country | Switzerland |

City | Zurich |

Period | 7/4/11 → 7/8/11 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Automata, Languages and Programming - 38th International Colloquium, ICALP 2011, Proceedings*(PART 1 ed., pp. 1-12). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6755 LNCS, No. PART 1). https://doi.org/10.1007/978-3-642-22006-7_1

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*Automata, Languages and Programming - 38th International Colloquium, ICALP 2011, Proceedings.*PART 1 edn, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), no. PART 1, vol. 6755 LNCS, pp. 1-12, 38th International Colloquium on Automata, Languages and Programming, ICALP 2011, Zurich, Switzerland, 7/4/11. https://doi.org/10.1007/978-3-642-22006-7_1

**Improved approximation for the directed spanner problem.** / Berman, Piotr; Bhattacharyya, Arnab; Makarychev, Konstantin; Raskhodnikova, Sofya; Yaroslavtsev, Grigory.

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

TY - GEN

T1 - Improved approximation for the directed spanner problem

AU - Berman, Piotr

AU - Bhattacharyya, Arnab

AU - Makarychev, Konstantin

AU - Raskhodnikova, Sofya

AU - Yaroslavtsev, Grigory

PY - 2011/7/11

Y1 - 2011/7/11

N2 - We give an -approximation algorithm for the problem of finding the sparsest spanner of a given directed graph G on n vertices. A spanner of a graph is a sparse subgraph that approximately preserves distances in the original graph. More precisely, given a graph G∈=∈(V,E) with nonnegative edge lengths d: E → ℝ≥0 and a stretch k ≥ 1, a subgraph H = (V,E H ) is a k-spanner of G if for every edge (u,v)ε E, the graph H contains a path from u to v of length at most k •d(u,v). The previous best approximation ratio was , due to Dinitz and Krauthgamer (STOC '11). We also present an improved algorithm for the important special case of directed 3-spanners with unit edge lengths. The approximation ratio of our algorithm is which almost matches the lower bound shown by Dinitz and Krauthgamer for the integrality gap of a natural linear programming relaxation. The best previously known algorithms for this problem, due to Berman, Raskhodnikova and Ruan (FSTTCS '10) and Dinitz and Krauthgamer, had approximation ratio .

AB - We give an -approximation algorithm for the problem of finding the sparsest spanner of a given directed graph G on n vertices. A spanner of a graph is a sparse subgraph that approximately preserves distances in the original graph. More precisely, given a graph G∈=∈(V,E) with nonnegative edge lengths d: E → ℝ≥0 and a stretch k ≥ 1, a subgraph H = (V,E H ) is a k-spanner of G if for every edge (u,v)ε E, the graph H contains a path from u to v of length at most k •d(u,v). The previous best approximation ratio was , due to Dinitz and Krauthgamer (STOC '11). We also present an improved algorithm for the important special case of directed 3-spanners with unit edge lengths. The approximation ratio of our algorithm is which almost matches the lower bound shown by Dinitz and Krauthgamer for the integrality gap of a natural linear programming relaxation. The best previously known algorithms for this problem, due to Berman, Raskhodnikova and Ruan (FSTTCS '10) and Dinitz and Krauthgamer, had approximation ratio .

UR - http://www.scopus.com/inward/record.url?scp=79960008212&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79960008212&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-22006-7_1

DO - 10.1007/978-3-642-22006-7_1

M3 - Conference contribution

AN - SCOPUS:79960008212

SN - 9783642220050

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

SP - 1

EP - 12

BT - Automata, Languages and Programming - 38th International Colloquium, ICALP 2011, Proceedings

ER -