## Abstract

For a positive parameter β, the β-bounded distance between a pair of vertices u,v in a weighted undirected graph G = (V,E,ω) is the length of the shortest u −v path in G with at most β edges, aka hops. For β as above and ϵ > 0, a (β,ϵ)-hopset of G = (V,E,ω) is a graph G_{H} = (V,H,ω_{H} ) on the same vertex set, such that all distances in G are (1 + ϵ)-approximated by β-bounded distances in G ∪ G_{H} . 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(n log 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 [11] 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 [11]. 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 |
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Title of host publication | SPAA 2019 - Proceedings of the 31st ACM Symposium on Parallelism in Algorithms and Architectures |

Publisher | Association for Computing Machinery |

Pages | 333-341 |

Number of pages | 9 |

ISBN (Electronic) | 9781450361842 |

DOIs | |

State | Published - 17 Jun 2019 |

Event | 31st ACM Symposium on Parallelism in Algorithms and Architectures, SPAA 2019 - Phoenix, United States Duration: 22 Jun 2019 → 24 Jun 2019 |

### Conference

Conference | 31st ACM Symposium on Parallelism in Algorithms and Architectures, SPAA 2019 |
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Country/Territory | United States |

City | Phoenix |

Period | 22/06/19 → 24/06/19 |

## Keywords

- Hopsets
- Shortest paths

## ASJC Scopus subject areas

- Software
- Theoretical Computer Science
- Hardware and Architecture