SIMPLE RANDOMIZED LEADER ELECTION WITH EXTENSIONS

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.