Fast perfect-information leader-election protocols with linear immunity

Abstract

In this paper we develop a leader election protocolP with the following features:

• 1. The protocol rums in theperfect information model: Every step taken by a player is visible to all others.

• 2. It haslinear immunity: IfP is run byn players and a coalition ofc ₁ n players deviates from the protocol, attempting to have one of them elected, their probability of success is <1-c ₂, wherec ₁,c ₂>0 are absolute constants.

• 3. It isfast: The running time ofP is polylogarithmic inn, the number of players.

A previous protocol by Alon and Naor achieving linear immunity in the perfect information model has a linear time complexity. The main ingredient of our protocol is areduction subprotocol. This is a way forn players to elect a subset of themselves which has the following property. Assume that up toen of the players are bad and try to have as many of them elected to the subset. Then with high probability, the fraction of bad players among the elected ones will not exceede in a significant way. The existence of such a reduction protocol is first established by a probabilistic argument. Later an explicit construction is provided which is based on the spectral properties of Ramanujan graphs.