On more variants of the Majority Problem

TL;DR

This paper investigates the minimum number of comparisons needed for a colorblind player to identify all green balls among red and green balls, considering cases with at most or exactly p green balls, extending known results and providing bounds.

Contribution

It extends the analysis of the Majority Problem by determining bounds and exact values for the number of comparisons needed in new scenarios with at most or exactly p green balls.

Findings

Derived bounds for Q(N,p,≤) and Q(N,p,=)

Extended known cases for the Majority Problem

Provided exact values for specific parameters

Abstract

The problem we are considering is the following. A colorblind player is given a set $B = \{b_1,b_2,...,b_N\}$ of $N$ colored balls. He knows that each ball is colored either red or green, and that there are less green balls (this will be called a Red-green coloring), but he cannot distinguish the two colors. For any two balls he can ask whether they are colored the same. His goal is to determine the set of all green balls of $B$ (and hence the set of all red balls). We study here the case where the Red-green coloring is such that there are at most $p$ green balls, where $p$ is given, and denote by $Q(N,p,\le)$ the minimum integer $k$ such that there exists a method that finds for sure, for any Red-green coloring, the color of each ball of $B$ after at most $k$ (color) comparisons. We extend the cases for which the exact value of $Q(N,p,\le)$ is known and provide lower and upper bounds as well as exact values for $Q(N, p, =)$ (defined similarly as $Q(N, p, \le)$, but for a Red-green coloring with exactly $p$ green balls).