Average-case analysis of some plurality algorithms

Abstract

Given a set of n elements, each of which is colored one of c colors, we must determine an element of the plurality (most frequently occurring) color by pairwise equal/unequal color comparisons of elements. We focus on the expected number of color comparisons when the c n colorings are equally probable. We analyze an obvious algorithm, showing that its expected performance is c 2 + c − 2/2 c n − O ( c 2 ), with variance Θ( c 2 n ). We present and analyze an algorithm for the case c = 3 colors whose average complexity on the 3 n equally probable inputs is 7083/5425 n + O (√ n ) = 1.3056… n + O (√ n ), substantially better than the expected complexity 5/3 n + O (1) = 1.6666… n + O (1) of the obvious algorithm. We describe a similar algorithm for c =4 colors whose average complexity on the 4 n equally probable inputs is 761311/402850 n + O (log n ) = 1.8898… n + O (log n ), substantially better than the expected complexity 9/4 n + O (1) = 2.25 n + O (1) of the obvious algorithm.