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The Singular Optimality of Distributed Computation in LOCAL

Authors Fabien Dufoulon , Gopal Pandurangan , Peter Robinson , Michele Scquizzato

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Author Details

Fabien Dufoulon
  • Lancaster University, UK
Gopal Pandurangan
  • University of Houston, TX, USA
Peter Robinson
  • Augusta University, GA, USA
Michele Scquizzato
  • University of Padova, Italy

Funding

  • Dufoulon, Fabien: Part of this work was done while visiting the University of Padova, partially supported by the "National Group for Scientific Computing" (GNCS-INdAM).
  • Pandurangan, Gopal: Supported in part by ARO Grant W911NF-231-0191 and NSF grant CCF-2402837.
  • Robinson, Peter: Supported in part by National Science Foundation (NSF) grant CCF-2402836.
  • Scquizzato, Michele: Supported in part by the Italian National Center for HPC, Big Data, and Quantum Computing.

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Fabien Dufoulon, Gopal Pandurangan, Peter Robinson, and Michele Scquizzato. The Singular Optimality of Distributed Computation in LOCAL. In 28th International Conference on Principles of Distributed Systems (OPODIS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 324, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.OPODIS.2024.26

Abstract

It has been shown that one can design distributed algorithms that are (nearly) singularly optimal, meaning they simultaneously achieve optimal time and message complexity (within polylogarithmic factors), for several fundamental global problems such as broadcast, leader election, and spanning tree construction, under the KT₀ assumption. With this assumption, nodes have initial knowledge only of themselves, not their neighbors. In this case the time and message lower bounds are Ω(D) and Ω(m), respectively, where D is the diameter of the network and m is the number of edges, and there exist (even) deterministic algorithms that simultaneously match these bounds.
On the other hand, under the KT₁ assumption, whereby each node has initial knowledge of itself and the identifiers of its neighbors, the situation is not clear. For the KT₁ CONGEST model (where messages are of small size), King, Kutten, and Thorup (KKT) showed that one can solve several fundamental global problems (with the notable exception of BFS tree construction) such as broadcast, leader election, and spanning tree construction with Õ(n) message complexity (n is the network size), which can be significantly smaller than m. Randomization is crucial in obtaining this result. While the message complexity of the KKT result is near-optimal, its time complexity is Õ(n) rounds, which is far from the standard lower bound of Ω(D). An important open question is whether one can achieve singular optimality for the above problems in the KT₁ CONGEST model, i.e., whether there exists an algorithm running in Õ(D) rounds and Õ(n) messages. Another important and related question is whether the fundamental BFS tree construction can be solved with Õ(n) messages (regardless of the number of rounds as long as it is polynomial in n) in KT₁. 
In this paper, we show that in the KT₁ LOCAL model (where message sizes are not restricted), singular optimality is achievable. Our main result is that all global problems, including BFS tree construction, can be solved in Õ(D) rounds and Õ(n) messages, where both bounds are optimal up to polylogarithmic factors. Moreover, we show that this can be achieved deterministically.

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
  • Mathematics of computing → Probabilistic algorithms
  • Mathematics of computing → Discrete mathematics
Keywords
  • Distributed algorithms
  • round and message complexity
  • BFS tree construction
  • leader election

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