An Almost Singularly Optimal Asynchronous Distributed MST Algorithm

Authors Fabien Dufoulon , Shay Kutten , William K. Moses Jr. , Gopal Pandurangan , David Peleg



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Author Details

Fabien Dufoulon
  • Department of Computer Science, University of Houston, Houston, TX, USA
Shay Kutten
  • Faculty of Industrial Engineering and Management, Technion - Israel Institute of Technology, Haifa, Israel
William K. Moses Jr.
  • Department of Computer Science, University of Houston, Houston, TX, USA
Gopal Pandurangan
  • Department of Computer Science, University of Houston, Houston, TX, USA
David Peleg
  • Department of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot, Israel

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Fabien Dufoulon, Shay Kutten, William K. Moses Jr., Gopal Pandurangan, and David Peleg. An Almost Singularly Optimal Asynchronous Distributed MST Algorithm. In 36th International Symposium on Distributed Computing (DISC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 246, pp. 19:1-19:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.DISC.2022.19

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, 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 Ω̃(D+√n) and Ω(m) are respective time and message lower bounds for distributed MST construction in the standard KT₀ 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 KT₁ 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
  • Mathematics of computing → Probabilistic algorithms
  • Mathematics of computing → Discrete mathematics
Keywords
  • Asynchronous networks
  • Minimum Spanning Tree
  • Distributed Algorithm
  • Singularly Optimal

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