## Abstract

For an undirected unweighted graph G = (V, E) with n vertices and m edges, let d(u, v) denote the distance from u ∈ V to v ∈ V in G. An (α, β)-stretch approximate distance oracle (ADO) for G is a data structure that given u, v ∈ V returns in constant (or near constant) time a value d̂(u, v) such that d(u, v) ≤ d̂(u, v) ≤ α · d(u, v) + β, for some reals α > 1, β. Thorup and Zwick [34] showed that one cannot beat stretch 3 with subquadratic space (in terms of n) for general graphs. Pǎtraşcu and Roditty [27] showed that one can obtain stretch 2 using O(m^{1}/^{3}n^{4}/^{3}) space, and so if m is subquadratic in n then the space usage is also subquadratic. Moreover, Pǎtraşcu and Roditty [27] showed that one cannot beat stretch 2 with subquadratic space even for graphs where m = Õ(n), based on the set-intersection hypothesis. In this paper we explore the conditions for which an ADO can beat stretch 2 while using subquadratic space. In particular, we show that if the maximum degree in G is ∆G ≤ O(n^{1}/k−^{ε}) for some 0 < ε ≤ 1/k, then there exists an ADO for G that uses Õ(n^{2− kε 3} ) space and has a (2, 1 − k)-stretch. For k = 2 this result implies a subquadratic sub-2 stretch ADO for graphs with ∆G ≤ O(n^{1}/^{2−ε}). Moreover, we prove a conditional lower bound, based on the set intersection hypothesis, which states that for any positive integer k ≤ log n, obtaining a sub-^{k+2}_{k} stretch for graphs with ∆G = Θ(n^{1/k}) requires Ω̃(n^{2}) space. Thus, for graphs with maximum degree Θ(n^{1}/^{2}), obtaining a sub-2 stretch requires Ω̃(n^{2}) space.

Original language | English |
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Title of host publication | 51st International Colloquium on Automata, Languages, and Programming, ICALP 2024 |

Editors | Karl Bringmann, Martin Grohe, Gabriele Puppis, Ola Svensson |

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

ISBN (Electronic) | 9783959773225 |

DOIs | |

State | Published - Jul 2024 |

Event | 51st International Colloquium on Automata, Languages, and Programming, ICALP 2024 - Tallinn, Estonia Duration: 8 Jul 2024 → 12 Jul 2024 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 297 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 51st International Colloquium on Automata, Languages, and Programming, ICALP 2024 |
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Country/Territory | Estonia |

City | Tallinn |

Period | 8/07/24 → 12/07/24 |

### Bibliographical note

Publisher Copyright:© Tsvi Kopelowitz, Ariel Korin, and Liam Roditty.

## Keywords

- Approximate distance oracle
- Graph algorithms
- data structures
- shortest path