 
Summary: Time and Message bounds for Election in Synchronous
and Asynchronous Complete Networks \Lambda
Yehuda Afek y Eli Gafni z
Abstract
This paper addresses the problem of distributively electing a leader in both syn
chronous and asynchronous complete networks. We present O(n log n) messages syn
chronous and asynchronous algorithms. The time complexity of the synchronous algo
rithm is O(log n), while that of the asynchronous algorithm is O(n). In the synchronous
case, we prove a lower bound of \Omega\Gamma n log n) on the message complexity. We also prove
that any messageoptimal synchronous algorithm
requires\Omega\Gammaqui n) time. In proving
these bounds we do not restrict the type of operations performed by nodes. The bounds
thus apply to general algorithms and not just to comparison based algorithms.
1 Introduction
In the election problem, a single node, called the leader, is to be selected from a set of nodes
which initially differ only in their identifiers (ids), with no node being aware of any other
id. An arbitrary subset of nodes wakes up spontaneously at arbitrary times and starts the
\Lambda A preliminary version of Section 3 was presented in the TwentySecond Annual Allerton Conference on
Communication, Control, and Computing, Allerton, IL October 35, 1984. A preliminary version of Section
4 was presented in the Fourth Annual ACM Symposium on Principles of Distributed Computing, Minaki,
