Average-case lower bounds for the plurality problem

• Authors: Laurent Alonso;Edward M. Reingold
• Venue: ACM Transactions on Algorithms (TALG)
• Year: 2008

Given a set of n elements, each of which is colored one of c ≥ 2 colors, we have to determine an element of the plurality (most frequently occurring) color by pairwise equal/unequal color comparisons of elements. We derive lower bounds for the expected number of color comparisons when the cn colorings are equally probable. We prove a general lower bound of c/3n − O(&sqrt;n) for c ≥ 2; we prove the stronger particular bounds of 7/6 n − O(&sqrt;n) for c = 3, 54/35n − O(&sqrt;n) for c = 4, 607/315n − O(&sqrt;n) for c = 5, 1592/693n − O(&sqrt;n) for c = 6, 7985/3003n − O(&sqrt;n) for c = 7, and 19402/6435n − O(&sqrt;n) for c = 8.