From Discrepancy to Majority

TL;DR

This paper presents an improved query method for identifying the majority color among n two-colored items using an oracle that reports subset discrepancies, along with a tighter lower bound on query complexity.

Contribution

It introduces a more efficient querying strategy and establishes a stronger lower bound for the problem of majority item selection via discrepancy queries.

Findings

Uses approximately n/⌊k/2⌋ queries, improving previous methods.

Provides a lower bound of n/(k-1)-O(n^{1/3}) queries, tighter than prior bounds.

Enhances understanding of query complexity in discrepancy-based majority detection.

Abstract

We show how to select an item with the majority color from $n$ two-colored items, given access to the items only through an oracle that returns the discrepancy of subsets of $k$ items. We use $n/\lfloor\tfrac{k}{2}\rfloor+O(k)$ queries, improving a previous method by De Marco and Kranakis that used $n-k+k^2/2$ queries. We also prove a lower bound of $n/(k-1)-O(n^{1/3})$ on the number of queries needed, improving a lower bound of $\lfloor n/k\rfloor$ by De Marco and Kranakis.