## Abstract

A singularly (near) optimal distributed algorithm is one that is (near) optimal in two criteria, namely, its time and message complexities. For synchronous CONGEST networks, such algorithms are known for fundamental distributed computing problems such as leader election [Kutten et al., JACM 2015] and Minimum Spanning Tree (MST) construction [Pandurangan et al., STOC 2017, Elkin, PODC 2017]. However, it is open whether a singularly (near) optimal bound can be obtained for the MST construction problem in general asynchronous CONGEST networks. In this paper, we present a randomized distributed MST algorithm that, with high probability, computes an MST in asynchronous CONGEST networks and takes Õ(D^{1+ε} + √n) time and Õ(m) messages^{1}, where n is the number of nodes, m the number of edges, D is the diameter of the network, and ε > 0 is an arbitrarily small constant (both time and message bounds hold with high probability). Since (Equation presented)(D + √n) and Ω(m) are respective time and message lower bounds for distributed MST construction in the standard KT_{0} model, our algorithm is message optimal (up to a polylog(n) factor) and almost time optimal (except for a D^{ε} factor). Our result answers an open question raised in Mashregi and King [DISC 2019] by giving the first known asynchronous MST algorithm that has sublinear time (for all D = O(n^{1-ε})) and uses Õ(m) messages. Using a result of Mashregi and King [DISC 2019], this also yields the first asynchronous MST algorithm that is sublinear in both time and messages in the KT_{1} CONGEST model. A key tool in our algorithm is the construction of a low diameter rooted spanning tree in asynchronous CONGEST that has depth Õ(D^{1+ε}) (for an arbitrarily small constant ε > 0) in Õ(D^{1+ε}) time and Õ(m) messages. To the best of our knowledge, this is the first such construction that is almost singularly optimal in the asynchronous setting. This tree construction may be of independent interest as it can also be used for efficiently performing basic tasks such as verified broadcast and convergecast in asynchronous networks.

Original language | English |
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Title of host publication | 36th International Symposium on Distributed Computing, DISC 2022 |

Editors | Christian Scheideler |

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

ISBN (Electronic) | 9783959772556 |

DOIs | |

State | Published - 1 Oct 2022 |

Externally published | Yes |

Event | 36th International Symposium on Distributed Computing, DISC 2022 - Augusta, United States Duration: 25 Oct 2022 → 27 Oct 2022 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 246 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 36th International Symposium on Distributed Computing, DISC 2022 |
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Country/Territory | United States |

City | Augusta |

Period | 25/10/22 → 27/10/22 |

### Bibliographical note

Publisher Copyright:© Fabien Dufoulon, Shay Kutten, William K. Moses Jr., Gopal Pandurangan, and David Peleg.

### Funding

Funding Fabien Dufoulon: This work was supported in part by NSF grants CCF-1717075, CCF-1540512, IIS-1633720, and BSF grant 2016419. Shay Kutten: This work was supported in part by the Bi-national Science Foundation (BSF) grant 2016419 and supported in part by ISF grant 1346/22. William K. Moses Jr.: This work was supported in part by NSF grants CCF1540512, IIS-1633720, CCF-1717075, and BSF grant 2016419. Gopal Pandurangan: This work was supported in part by NSF grants CCF-1717075, CCF-1540512, IIS-1633720, and BSF grant 2016419. David Peleg: This work was supported in part by the US-Israel Binational Science Foundation grant 2018043.

Funders | Funder number |
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National Science Foundation | 2016419, CCF-1540512, IIS-1633720, CCF-1717075 |

Iowa Science Foundation | 1346/22 |

United States-Israel Binational Science Foundation | 2018043 |

## Keywords

- Asynchronous networks
- Distributed Algorithm
- Minimum Spanning Tree
- Singularly Optimal