Randomized algorithms for finding a majority element

TL;DR

This paper introduces a randomized algorithm for identifying a majority element among colored balls, reducing the number of comparisons needed compared to deterministic methods, with proven bounds on efficiency.

Contribution

It presents a novel randomized algorithm that finds a majority element with fewer comparisons and establishes lower bounds on the expected comparisons needed.

Findings

Randomized algorithm uses approximately 7n/6 comparisons with high probability.

Expected comparisons for any randomized method are at least 1.019n.

Deterministic methods require at least 3n/2 comparisons in worst case.

Abstract

Given $n$ colored balls, we want to detect if more than $\lfloor n/2\rfloor$ of them have the same color, and if so find one ball with such majority color. We are only allowed to choose two balls and compare their colors, and the goal is to minimize the total number of such operations. A well-known exercise is to show how to find such a ball with only $2n$ comparisons while using only a logarithmic number of bits for bookkeeping. The resulting algorithm is called the Boyer--Moore majority vote algorithm. It is known that any deterministic method needs $\lceil 3n/2\rceil-2$ comparisons in the worst case, and this is tight. However, it is not clear what is the required number of comparisons if we allow randomization. We construct a randomized algorithm which always correctly finds a ball of the majority color (or detects that there is none) using, with high probability, only $7n/6+o(n)$ comparisons. We also prove that the expected number of comparisons used by any such randomized method is at least $1.019n$.