## Abstract

Thorup and Zwick (J. ACM 52(1):1-24, 2005 and STOC'01) in their seminal work introduced the notion of distance oracles. Given an n-vertex weighted undirected graph with m edges, they show that for any integer k≥1 it is possible to preprocess the graph in Õ(mn^{1/k}) time and generate a compact data structure of size O(kn ^{1+1/k} ). For each pair of vertices, it is then possible to retrieve an estimated distance with multiplicative stretch 2k-1 in O(k) time. For k=2 this gives an oracle of O(n ^{1.5}) size that produces in constant time estimated distances with stretch 3. Recently, Pǎtraşcu and Roditty (In: Proc. of 51st FOCS, 2010) broke the theoretical status-quo in the field of distance oracles and obtained a distance oracle for sparse unweighted graphs of O(n ^{5/3}) size that produces in constant time estimated distances with stretch 2. In this paper we show that it is possible to break the stretch 2 barrier at the price of non-constant query time in unweighted undirected graphs. We present a data structure that produces estimated distances with 1+ε stretch. The size of the data structure is O(nm ^{1-ε′}) and the query time is Õ(mn^{1-ε′}). Using it for sparse unweighted graphs we can get a data structure of size O(n ^{1.87}) that can supply in O(n ^{0.87}) time estimated distances with multiplicative stretch 1.75.

Original language | English |
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Pages (from-to) | 516-528 |

Number of pages | 13 |

Journal | Algorithmica |

Volume | 67 |

Issue number | 4 |

DOIs | |

State | Published - Dec 2013 |

### Bibliographical note

Funding Information:L. Roditty’s work is supported by Israel Science Foundation (grant no. 822/10).

## Keywords

- Data structures
- Distance oracles
- Graph algorithms