## Abstract

Given an undirected possibly weighted n-vertex graph G = (V,E) and a set P V 2 of pairs, a subgraph S = (V,E) is called a P-pairwise α-spanner of G, if for every pair (u, v) P we have dS(u, v) α dG(u, v). The parameter α is called the stretch of the spanner, and its size overhead is define as |E| |P| . A surprising connection was recently discussed between the additive stretch of (1+, β)-spanners, to the hopbound of (1 + , β)-hopsets. A long sequence of works showed that if the spanner/hopset has size n1+1/k for some parameter k 1, then β 1 log k. In this paper we establish a new connection to the size overhead of pairwise spanners. In particular, we show that if |P| n1+1/k, then a P-pairwise (1 + )-spanner must have size at least β |P| with β 1 log k (a near matching upper bound was recently shown in [18]). That is, the size overhead of pairwise spanners has similar bounds to the hopbound of hopsets, and to the additive stretch of spanners. We also extend the connection between pairwise spanners and hopsets to the large stretch regime, by showing nearly matching upper and lower bounds for P-pairwise α-spanners. In particular, we show that if |P| n1+1/k, then the size overhead is β k α. A source-wise spanner is a special type of pairwise spanner, for which P = A × V for some A V . A prioritized spanner is given also a ranking of the vertices V = (v1, . . . , vn), and is required to provide improved stretch for pairs containing higher ranked vertices. By using a sequence of reductions: from pairwise spanners to source-wise spanners to prioritized spanners, we improve on the state-of-The-Art results for source-wise and prioritized spanners. Since our spanners can be equipped with a path-reporting mechanism, we also substantially improve the known bounds for path-reporting prioritized distance oracles. Specifically, we provide a path-reporting distance oracle, with size O(n (log log n)2), that has a constant stretch for any query that contains a vertex ranked among the first n1 vertices (for any constant 0). Such a result was known before only for non-path-reporting distance oracles.

Original language | American English |
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Title of host publication | 15th Innovations in Theoretical Computer Science Conference, ITCS 2024 |

Editors | Venkatesan Guruswami |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959773096 |

DOIs | |

State | Published - 1 Jan 2024 |

Event | 15th Innovations in Theoretical Computer Science Conference, ITCS 2024 - Berkeley, United States Duration: 30 Jan 2024 → 2 Feb 2024 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 287 |

### Conference

Conference | 15th Innovations in Theoretical Computer Science Conference, ITCS 2024 |
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Country/Territory | United States |

City | Berkeley |

Period | 30/01/24 → 2/02/24 |

## Keywords

- Graph Algorithms
- Shortest Paths
- Spanners

## All Science Journal Classification (ASJC) codes

- Software