## Abstract

Consider an undirected weighted graph *G* = (*V, E*, ω. We study the problem of computing (1 + ϵ)approximate shortest paths for *S × V* , for a subset *S* ⊆ *V* of |*S*| = *n ^{r}* sources, for some 0 < r ≤ 1. We devise a significantly improved algorithm for this problem in the entire range of parameter r, in both the classical centralized and the parallel (PRAM) models of computation, and in a wide range of r in the distributed (Congested Clique) model. Specifically, our centralized algorithm for this problem requires time Õ(|

*E*| · n

^{o(1)}+ n

^{ω}(r)), where n

^{ω}(r) is the time required to multiply an n

^{r}× n matrix by an n × n one. Our PRAM algorithm has polylogarithmic time (log n)

^{O}(1/ρ), and its work complexity is Õ(|

*E*| · n

^{ρ}+ n

^{ω}(r)), for any arbitrarily small constant ρ > 0. In particular, for r ≤ 0.313 . . ., our centralized algorithm computes S × V (1 + ϵ)-approximate shortest paths in n

^{2+}o

^{(1)}time. Our PRAM polylogarithmic-time algorithm has work complexity O(|E| · n

^{ρ}+ n

^{2+}o

^{(1)}), for any arbitrarily small constant ρ > 0. Previously existing solutions either require centralized time/parallel work of O(|E| · |S|) or provide much weaker approximation guarantees. In the Congested Clique model, our algorithm solves the problem in polylogarithmic time for |S| = n

^{r}sources, for r ≤ 0.655, while previous state-of-the-art algorithms did so only for

*r*≤ 1/2. Moreover, it improves previous bounds for all

*r*> 1/2. For unweighted graphs, the running time is improved further to poly(log log n) for

*r*≤ 0.655. Previously this running time was known for

*r*≤ 1/2.

Original language | American English |
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Title of host publication | 39^{th} International Symposium on Theoretical Aspects of Computer Science, STACS 2022 |

Editors | Petra Berenbrink, Benjamin Monmege |

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

Pages | 27:1-27:22 |

Volume | 219 |

ISBN (Electronic) | 9783959772228 |

DOIs | |

State | Published - 9 Mar 2022 |

Event | 39th International Symposium on Theoretical Aspects of Computer Science, STACS 2022 - Virtual, Marseille, France Duration: 15 May 2022 → 18 May 2022 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 219 |

### Conference

Conference | 39th International Symposium on Theoretical Aspects of Computer Science, STACS 2022 |
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Country/Territory | France |

City | Virtual, Marseille |

Period | 15/05/22 → 18/05/22 |

## Keywords

- Hopsets
- Matrix multiplication
- Shortest paths

## All Science Journal Classification (ASJC) codes

- Software