# On the computational power needed to elect a leader

## Abstract

Consider a ring of identical processors that wish to elect a leader. Itai and Rodeh have shown that if *n*, the size of the ring, is not known then there exists no algorithm in which the processors can sense termination. However, for all *ε*>0 there exist leader election algorithms which terminate when all messages arrive. These algorithms elect a leader with error probability ≤ *ε*. They and most subsequent authors have concentrated on the message complexity, and have disregarded the amount of local memory required.

Here we consider a ring in which the amount of local memory in each processor is bounded by a number which is independent of the size of the ring, and depends only on the allowed error rate. We present three algorithms, one in which the probability of error is *O*(1/*n*^{ α }), the memory is *O*(log *α*) bits and the communication complexity is *O*(*n*^{2} log *n*) bits, where *α*>1 is an arbitrary parameter. In the second algorithm, for each *ε*>0 the probability of error is ≤ *ε*, *O*(logloglog(1/*ε*)) bits of memory are required and the communication complexity is *O*((*n/ε*)log *n* log(1/*ε*)(loglog *n*+log(1/*ε*))) bits. The third algorithm always terminates though its communication complexity may be larger. Since the computation for each message is at most exponential in the number of bits of local memory, the amount of computation is also independent of the size of the ring.

These results are extended to additional topologies.

## Keywords

Basic Algorithm Local Memory Communication Complexity Finite Automaton Leader State## Preview

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