## Abstract

Hopsets are a fundamental graph-theoretic and graph-algorithmic construct, and they are widely used for distance-related problems in a variety of computational settings. Currently existing constructions of hopsets produce hopsets either with Ω (nlog n) edges, or with a hopbound n^{Ω (1)}. In this paper we devise a construction of linear-size hopsets with hopbound (ignoring the dependence on ϵ) (log log n) ^{log}^{log}^{n}^{+}^{O}^{(}^{1}^{)}. This improves the previous hopbound for linear-size hopsets almost exponentially. We also devise efficient implementations of our construction in PRAM and distributed settings. The only existing PRAM algorithm [19] for computing hopsets with a constant (i.e., independent of n) hopbound requires n^{Ω (1)} time. We devise a PRAM algorithm with polylogarithmic running time for computing hopsets with a constant hopbound, i.e., our running time is exponentially better than the previous one. Moreover, these hopsets are also significantly sparser than their counterparts from [19]. We apply these hopsets to achieve the following online variant of shortest paths in the PRAM model: preprocess a given weighted graph within polylogarithmic time, and then given any query vertex v, report all approximate shortest paths from v in constant time. All previous constructions of hopsets require either polylogarithmic time per query or polynomial preprocessing time.

Original language | English |
---|---|

Pages (from-to) | 419-437 |

Number of pages | 19 |

Journal | Distributed Computing |

Volume | 35 |

Issue number | 5 |

DOIs | |

State | Published - 1 Oct 2022 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- Hardware and Architecture
- Computer Networks and Communications
- Computational Theory and Mathematics