Analysis of Boyer and Moore's MJRTY algorithm

Given a set {x1, x2, . . . , xn}, each element of which is colored either red or blue, we must determine an element of the majority color by making equal/not equal color comparisons xu : xv; when n is even, we must report that there is no majority if there are equal numbers of each color. How many such questions are necessary and sufficient? It is easy to obtain an algorithm using at most n−ν(n) questions, where ν(n) denotes the number of 1-bits in the binary representation of n; furthermore, n− ν(n) is a lower bound on the number of questions needed (see [5] and [1]). In [2], the average case was investigated: Assuming all 2 distinct colorings of the n elements are equally probable, 2n 3 − √