New lower bounds for distributed leader finding in asynchronous rings of processors

Abstract

Consider a distributed system of n processors, arranged in a ring. All processor are labeled with a unique indentity-number, but are otherwise identical. In this paper we proof some new lowerbounds for the problem of determining a leader (e.g. the largest number processor). This selected processor then can act as central controller (coordinator) e.g. in a restart-procedure after a crash of the distributed system.

We show that the average number of messages needed for leader-finding in bidirectional rings with the ring size not known to the processors is at least 1/2nH _n (H _n the n’th Harmonic number). We give several lower bounds for the case that the ringsize is known in advance to the processors, for algorithms that use only comparisons between identities. For arbitrary algorithms on unidirectional rings with known ring size, we show a worst-case lowerbound of (1 —∈) nH _n messages. Also, using results from extremal graph theory, we give an easy proof for an 1/4 n log n — O(n)lowerbound for the average number of messages on unidirectional rings with known ringsize n, for n a power of 2, where the indentities may be chosen from a set I, with size as small as 2n. Similar Ω(n log n) lowerbounds can be proved for the bidirectional case, arbitrary n, and |I| > c • n, for any constant c > 1.

This research was done, while the author was visiting the Laboratory for Computer Science of the Massachusetts Institute of Technology, with financial support by the Netherlands Organization for the Advancement of Pure Research (Z.W.O.).