## Abstract

Given an undirected weighted graph, an (approximate) distance oracle is a data structure that can (approximately) answer distance queries. A Path-Reporting Distance Oracle, or PRDO, is a distance oracle that must also return a path between the queried vertices. Given a graph on n vertices and an integer parameter k ≥ 1, Thorup and Zwick [22] showed a PRDO with stretch 2k − 1, size O(k · n^{1+1/k}) and query time O(k) (for the query time of PRDOs, we omit the time needed to report the path itself). Subsequent works [20, 7, 8] improved the size to O(n^{1+1/k}) and the query time to O(1). However, these improvements produce distance oracles which are not path-reporting. Several other works [12, 13] focused on small size PRDO for general graphs, but all known results on distance oracles with linear size suffer from polynomial stretch, polynomial query time, or not being path-reporting. In this paper we devise the first linear size PRDO with poly-logarithmic stretch and low query time O(log log n). More generally, for any integer k ≥ 1, we obtain a PRDO with stretch at most O(k^{4.82}), size O(n^{1+1/k}), and query time O(log k). In addition, we can make the size of our PRDO as small as n + o(n), at the cost of increasing the query time to poly-logarithmic. For unweighted graphs, we improve the stretch to O(k^{2}). We also consider pairwise PRDO, which is a PRDO that is only required to answer queries from a given set of pairs P. An exact PRDO of size O(n + |P|^{2}) and constant query time was provided in [13]. In this work we dramatically improve the size, at the cost of slightly increasing the stretch. Specifically, given any ϵ > 0, we devise a pairwise PRDO with stretch 1 + ϵ, constant query time, and near optimal size n^{o}^{(1)} · (n + |P|).

Original language | English |
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Title of host publication | 19th Scandinavian Symposium on Algorithm Theory, SWAT 2024 |

Editors | Hans L. Bodlaender |

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

ISBN (Electronic) | 9783959773188 |

DOIs | |

State | Published - 1 Jun 2024 |

Event | 19th Scandinavian Symposium on Algorithm Theory, SWAT 2024 - Helsinki, Finland Duration: 12 Jun 2024 → 14 Jun 2024 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 294 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 19th Scandinavian Symposium on Algorithm Theory, SWAT 2024 |
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Country/Territory | Finland |

City | Helsinki |

Period | 12/06/24 → 14/06/24 |

## Keywords

- Distance Oracles
- Graph Algorithms
- Shortest Paths

## ASJC Scopus subject areas

- Software