The space complexity of the leader election in anonymous networks

It is known that the leader election in anonymous networks is not always solvable, and solvable/unsolvable cases are characterized by the network topologies. Therefore a distributed leader election algorithm is required to elect a leader when it is possible, otherwise recognize the impossibility and stop. Although former studies proposed several leader election algorithms, the space complexity of the problem is not considered well. This paper focuses on the space complexity, that is, the necessary or sufficient number of bits on processors to execute a leader election algorithm. First we show that only one bit memory is sufficient for a leader election algorithm which is specific to a fixed n. We then show that a general algorithm can solve the leader election for arbitrary n if each processor has O(n log d) bits memory where d is the maximum degree of a processor. Finally, we give a lower bound Ω(log n) on the space complexity, that is, we show that it is impossible to construct a leader election algorithm if only log n bits are available for a processor.