Finding a non-minority ball with majority answers

TL;DR

This paper investigates the minimum number of queries needed to identify a ball with a majority or non-minority color among n balls, using majority responses from triple queries, with results differing for adaptive and non-adaptive strategies.

Contribution

It establishes tight bounds on the query complexity for finding a non-minority ball in both adaptive and non-adaptive settings.

Findings

Adaptive queries require Θ(n) queries.

Non-adaptive queries require Θ(n^3) queries.

Results extend to related problems in majority identification.

Abstract

Suppose we are given a set of $n$ balls $\{b_1,\ldots,b_n\}$ each colored either red or blue in some way unknown to us. To find out some information about the colors, we can query any triple of balls $\{b_{i_1},b_{i_2},b_{i_3}\}$. As an answer to such a query we obtain (the index of) a {\em majority ball}, that is, a ball whose color is the same as the color of another ball from the triple. Our goal is to find a {\em non-minority ball}, that is, a ball whose color occurs at least $\frac n2$ times among the $n$ balls. We show that the minimum number of queries needed to solve this problem is $\Theta(n)$ in the adaptive case and $\Theta(n^3)$ in the non-adaptive case. We also consider some related problems.