## Abstract

Let G = (V, E) be a weighted undirected graph with n vertices and m edges, and let dG(u, v) be the length of the shortest path between u and v in G. In this paper we present a unified approach for obtaining algorithms for all pairs approximate shortest paths in weighted undirected graphs. For every integer k ≥ 2 we show that there is an Õ(n^{2} + kn^{2}−3^{/k}m^{2/k}) expected running time algorithm that computes a matrix M such that for every u, v ∈ V : (equation presented) Previous algorithms obtained only specific approximation factors. Baswana and Kavitha [FOCS 2006, SICOMP 2010] presented a 2-approximation algorithm with expected running time of Õ(n^{2} + m√n) and a 7/3-approximation algorithm with expected running time of Õ(n^{2} + m^{2}/^{3}n). Their results improved upon the results of Cohen and Zwick [SODA 1997, JoA 2001] for graphs with m = o(n^{2}). Kavitha [FSTTCS 2007, Algorithmica 2012] presented a 5/2-approximation algorithm with expected running time of Õ(n^{9}/^{4}). For k = 2 and k = 3 our result gives the algorithms of Baswana and Kavitha. For k = 4, we 1 get a 5/2-approximation algorithm with Õ(n^{54} m2) expected running time. This improves upon the running time of Õ(n^{9}/^{4}) due to Kavitha, when m = o(n^{2}). Our unified approach reveals that all previous algorithms are a part of a family of algorithms that exhibit a smooth tradeoff between approximation of 2 and 3, and are not sporadic unrelated results. Moreover, our new algorithm uses, among other ideas, the celebrated approximate distance oracles of Thorup and Zwick [STOC 2001, JACM 2005] in a non standard way, which we believe is of independent interest, due to their extensive usage in a variety of applications.

Original language | English |
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Title of host publication | 29th Annual European Symposium on Algorithms, ESA 2021 |

Editors | Petra Mutzel, Rasmus Pagh, Grzegorz Herman |

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

ISBN (Electronic) | 9783959772044 |

DOIs | |

State | Published - 1 Sep 2021 |

Event | 29th Annual European Symposium on Algorithms, ESA 2021 - Vitual, Lisbon, Portugal Duration: 6 Sep 2021 → 8 Sep 2021 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 204 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 29th Annual European Symposium on Algorithms, ESA 2021 |
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Country/Territory | Portugal |

City | Vitual, Lisbon |

Period | 6/09/21 → 8/09/21 |

### Bibliographical note

Publisher Copyright:© Maor Akav and Liam Roditty; licensed under Creative Commons License CC-BY 4.0

## Keywords

- Approximate all pairs of shortest paths
- Distance oracles
- Graph algorithms