Majority problems of large query size

TL;DR

This paper investigates efficient querying strategies to identify a majority color among balls with unknown coloring, improving bounds on the number of queries needed in two different models of the problem.

Contribution

The paper refines the bounds on the minimum number of queries required for majority detection in large query size models, advancing previous results by De Marco and Kranakis.

Findings

Improved bounds for the number of queries in the General Model.

Enhanced bounds for the Counting Model.

Better understanding of query complexity in majority problems.

Abstract

We study two models of the Majority problem. We are given n balls and an unknown coloring of them with two colors. We can ask sets of balls of size k as queries, and in the so-called General Model the answer to a query shows if all the balls in the set are of the same color or not. In the so-called Counting Model the answer to a query gives the difference between the cardinalities of the color classes in the query. Our goal is to show a ball of the larger color class, or prove that the color classes are of the same size, using as few queries as possible. In this paper we improve the bounds given by De Marco and Kranakis for the number of queries needed.