## Abstract

Let G = (V, E) be an unweighted undirected graph with n vertices and m edges. Dor, Halperin, and Zwick [FOCS 1996, SICOMP 2000] presented an Õ(n^{2})-time algorithm that computes estimated distances with a multiplicative approximation of 3. Berman and Kasiviswanathan [WADS 2007] improved the approximation of Dor et al. and presented an Õ(n^{2})time algorithm that produces for every u, v ∈ V an estimate (Equation presented)(u, v) such that: d_{G}(u, v) ≤ (Equation presented)(u, v) ≤ 2d_{G}(u, v) + 1. We refer to such an approximation as an (α, β)-approximation, where α is the multiplicative approximation and β is the additive approximation. A prerequisite for an O(n^{2}−^{ε})-time algorithm, where ε ∈ (0, 1), is a data structure that uses O(n^{2−δ}) space, for some δ ≥ ε, and answers queries in constant time. An O(n^{2}−^{ε})-time (3, 0)-approximation algorithm became plausible after Thorup and Zwick [STOC 2001, JACM 2005] presented their approximate distance oracles, and in particular an O(n^{1.5})-space data structure that reports a (3, 0)-approximate distance in O(1) time. Indeed, using Thorup and Zwick distance oracles together with more ideas, Baswana, Gaur, Sen, and Upadhyay [ICALP 2008] improved the running time of Dor et al., and obtained an O(n^{2}−^{ε}) time algorithm, at the cost of introducing also an additive approximation. They presented an algorithm that in Õ(m+ n^{23}/^{12}) expected running time constructs an O(n^{1.5})-space data structure, that in O(1) time reports a (3, 14)-approximate distance. An O(n^{2}−^{ε})-time (2, 1)-approximation algorithm became plausible only after Pǎtraşcu and Roditty [FOCS 2010, SICOMP 2014] presented an O(n^{5}/^{3})-space data structure that reports (2, 1)-approximate distances in O(1) time. However, only few years ago, Sommer [ICALP 2016] obtained an Õ(n^{2}) time algorithm that computes a (2, 1)-distance oracle with Õ(n^{5}/^{3}) space. This leads to the following natural question of whether Ω(n^{2}) time is a lower bound for any (3−α, β)-approximation, where α ∈ (0, 1), and β is constant. In this paper we show that this is not the case by presenting an algorithm that for every ε ∈ (0, 1/2) computes in Õ(m) + n^{2}−Ω time an Õ(n^{1 56} )-space data structure that in O(1/ε) time reports, for every u, v ∈ V , an estimate (Equation presented)(u, v) such that: (Equation presented). Our result improves, simultaneously, the running time and the multiplicative approximation of the Õ(n^{2})-time (3, 0)-approximation algorithm of Dor et al. at the cost of introducing also an additive approximation.

Original language | English |
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Title of host publication | 31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020 |

Editors | Shuchi Chawla |

Publisher | Association for Computing Machinery |

Pages | 1-11 |

Number of pages | 11 |

ISBN (Electronic) | 9781611975994 |

State | Published - 2020 |

Event | 31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020 - Salt Lake City, United States Duration: 5 Jan 2020 → 8 Jan 2020 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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Volume | 2020-January |

### Conference

Conference | 31st Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2020 |
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Country/Territory | United States |

City | Salt Lake City |

Period | 5/01/20 → 8/01/20 |

### Bibliographical note

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