SIMPLE RANDOMIZED LEADER ELECTION WITH EXTENSIONS
Date
1990-10-01
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Abstract
Leader election causes a unique processor to be
distinguished from among a collection of processors. As the
study of this problem progressed, increasingly efficient,
then more general algorithms emerged. Eventually, Las Vegas
algorithms for leader election on rings with identifiers and
for those without that achieve asymptotically optimal expected
message and bit communication complexity emerged (AAGHK).
In this paper, the same results are achieved with a much simpler algorithm
than previously proposed. That is, on a ring of size $n$ without
identifiers where an $N$ is known that satisfies $N~<=~n~<~2N~-~1$, a
leader is elected using $0(n log n)$ expected bits. If distinct
identifiers are available, then the algorithm can be adapted to use
$0(nm)$ expected bits even without knowledge of the ring size, where
$m$ is the size of the longest identifier. These results are
optimal in communication complexity and in generality.
The algorithm's simplicity facilitates not only its proof of correctness,
but also its extension to several other problems. An optimal algorithm for
ring orientation follows easily even for situations where deterministic
orientation is impossible. The algorithm also generalizes to an optimal
(expected bit complexity $0(n))$ Las Vegas algorithm for election in an
oriented complete graph. This algorithm, in turn, is adapted to an election
algorithm in an oriented sparse graph with no degradation in communication
complexity.
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Computer Science