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Singularly Optimal Randomized Leader Election

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

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Author Details

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

Funding

  • Kutten, Shay: This work was supported in part by the Bi-national Science Foundation (BSF) grant 2016419.
  • Moses Jr., William K.: This work was supported in part by the BSF grant 2016419 and in part by a Technion fellowship.
  • Pandurangan, Gopal: G. Pandurangan was supported, in part, by NSF grants CCF-1527867, CCF-1540512, IIS-1633720, CCF-1717075, and BSF grant 2016419.
  • Peleg, David: Supported in part by the US-Israel Binational Science Foundation grant 2016732.

Cite As Get BibTex

Shay Kutten, William K. Moses Jr., Gopal Pandurangan, and David Peleg. Singularly Optimal Randomized Leader Election. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 22:1-22:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.DISC.2020.22

Abstract

This paper concerns designing distributed algorithms that are singularly optimal, i.e., algorithms that are simultaneously time and message optimal, for the fundamental leader election problem in networks. Our main result is a randomized distributed leader election algorithm for asynchronous complete networks that is essentially (up to a polylogarithmic factor) singularly optimal. Our algorithm uses O(n) messages with high probability and runs in O(log² n) time (with high probability) to elect a unique leader. The O(n) message complexity should be contrasted with the Ω(n log n) lower bounds for the deterministic message complexity of leader election algorithms (regardless of time), proven by Korach, Moran, and Zaks (TCS, 1989) for asynchronous algorithms and by Afek and Gafni (SIAM J. Comput., 1991) for synchronous networks. Hence, our result also separates the message complexities of randomized and deterministic leader election. More importantly, our (randomized) time complexity of O(log² n) for obtaining the optimal O(n) message complexity is significantly smaller than the long-standing Θ̃(n) time complexity obtained by Afek and Gafni and by Singh (SIAM J. Comput., 1997) for message optimal (deterministic) election in asynchronous networks. Afek and Gafni also conjectured that Θ̃(n) time would be optimal for message-optimal asynchronous algorithms. Our result shows that randomized algorithms are significantly faster. 
Turning to synchronous complete networks, Afek and Gafni showed an essentially singularly optimal deterministic algorithm with O(log n) time and O(n log n) messages. Ramanathan et al. (Distrib. Comput. 2007) used randomization to improve the message complexity, and showed a randomized algorithm with O(n) messages but still with O(log n) time (with failure probability O(1 / log^{Ω(1)}n)). Our second result shows that synchronous complete networks admit a tightly singularly optimal randomized algorithm, with O(1) time and O(n) messages (both bounds are optimal). Moreover, our algorithm’s time bound holds with certainty, and its message bound holds with high probability, i.e., 1-1/n^c for constant c. 
Our results demonstrate that leader election can be solved in a simultaneously message and time-efficient manner in asynchronous complete networks using randomization. It is open whether this is possible in asynchronous general networks.

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
  • Mathematics of computing → Probabilistic algorithms
  • Mathematics of computing → Discrete mathematics
Keywords
  • Leader election
  • Asynchronous systems
  • Randomized algorithms
  • Singularly optimal
  • Complete networks

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