## Abstract

An additive + β spanner of a graph G is a subgraph which preserves distances up to an additive + β error. Additive spanners are well-studied in unweighted graphs but have only recently received attention in weighted graphs [Elkin et al. 2019 and 2020, Ahmed et al. 2020]. This paper makes two new contributions to the theory of weighted additive spanners. For weighted graphs, [Ahmed et al. 2020] provided constructions of sparse spanners with global error β= cW, where W is the maximum edge weight in G and c is constant. We improve these to local error by giving spanners with additive error + cW(s, t) for each vertex pair (s, t), where W(s, t) is the maximum edge weight along the shortest s–t path in G. These include pairwise + (2 + ε) W(·, · ) and + (6 + ε) W(·, · ) spanners over vertex pairs P⊆ V× V on O_{ε}(n| P|^{1 / 3}) and O_{ε}(n| P|^{1 / 4}) edges for all ε> 0, which extend previously known unweighted results up to ε dependence, as well as an all-pairs + 4 W(·, · ) spanner on O~ (n^{7 / 5}) edges. Besides sparsity, another natural way to measure the quality of a spanner in weighted graphs is by its lightness, defined as the total edge weight of the spanner divided by the weight of an MST of G. We provide a + εW(·, · ) spanner with O_{ε}(n) lightness, and a + (4 + ε) W(·, · ) spanner with O_{ε}(n^{2 / 3}) lightness. These are the first known additive spanners with nontrivial lightness guarantees. All of the above spanners can be constructed in polynomial time.

Original language | English |
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Title of host publication | Graph-Theoretic Concepts in Computer Science - 47th International Workshop, WG 2021, Revised Selected Papers |

Editors | Lukasz Kowalik, Michal Pilipczuk, Pawel Rzazewski |

Publisher | Springer Science and Business Media Deutschland GmbH |

Pages | 361-373 |

Number of pages | 13 |

ISBN (Print) | 9783030868376 |

DOIs | |

State | Published - 1 Jan 2021 |

Externally published | Yes |

Event | 47th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2021 - Virtual, Online Duration: 23 Jun 2021 → 25 Jun 2021 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 12911 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 47th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2021 |
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City | Virtual, Online |

Period | 23/06/21 → 25/06/21 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- General Computer Science