Chapter

Distributed Computing and Networking

Volume 7730 of the series Lecture Notes in Computer Science pp 348-362

Sublinear Bounds for Randomized Leader Election

  • Shay KuttenAffiliated withInformation Systems Group, Faculty of Industrial Engineering and Management, Technion - Israel Institute of Technology
  • , Gopal PanduranganAffiliated withDivision of Mathematical Sciences, Nanyang Technological UniversityDepartment of Computer Science, Brown University
  • , David PelegAffiliated withDepartment of Computer Science and Applied Mathematics, Weizmann Institute of Science
  • , Peter RobinsonAffiliated withDivision of Mathematical Sciences, Nanyang Technological University
  • , Amitabh TrehanAffiliated withInformation Systems Group, Faculty of Industrial Engineering and Management, Technion - Israel Institute of Technology

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Abstract

This paper concerns randomized leader election in synchronous distributed networks. A distributed leader election algorithm is presented for complete n-node networks that runs in O(1) rounds and (with high probability) takes only \(O(\sqrt{n}\log^{3/2} n)\) messages to elect a unique leader (with high probability). This algorithm is then extended to solve leader election on any connected non-bipartite n-node graph G in O(τ(G)) time and \(O(\tau(G)\sqrt{n}\log^{3/2} n)\) messages, where τ(G) is the mixing time of a random walk on G. The above result implies highly efficient (sublinear running time and messages) leader election algorithms for networks with small mixing times, such as expanders and hypercubes. In contrast, previous leader election algorithms had at least linear message complexity even in complete graphs. Moreover, super-linear message lower bounds are known for time-efficient deterministic leader election algorithms. Finally, an almost-tight lower bound is presented for randomized leader election, showing that \(\Omega(\sqrt n)\) messages are needed for any O(1) time leader election algorithm which succeeds with high probability. It is also shown that Ω(n 1/3) messages are needed by any leader election algorithm that succeeds with high probability, regardless of the number of the rounds. We view our results as a step towards understanding the randomized complexity of leader election in distributed networks.