Finding a solution to the Erd\H{o}s-Ginzburg-Ziv theorem in $O(n\log\log\log n)$ time

TL;DR

This paper introduces faster algorithms for the Erd ext{"o}s-Ginzburg-Ziv theorem, achieving $O(n ext{log} ext{log} ext{log} n)$ time, significantly improving over previous methods and demonstrating a new application of boolean convolution.

Contribution

The paper presents a novel $O(n ext{log} ext{log} ext{log} n)$ algorithm for the theorem, surpassing the traditional $O(n ext{log} n)$ approaches and showcasing a faster boolean convolution implementation.

Findings

Achieved $O(n ext{log} ext{log} ext{log} n)$ time complexity.

Provided practical and theoretical algorithms.

Improved the computational efficiency for the theorem.

Abstract

The Erd\H{o}s-Ginzburg-Ziv theorem states that for any sequence of $2n-1$ integers, there exists a subsequence of $n$ elements whose sum is divisible by $n$. In this article, we provide a simple, practical $O(n\log\log n)$ algorithm and a theoretical $O(n\log\log\log n)$ algorithm, both of which improve upon the best previously known $O(n\log n)$ approach. This shows that a specific variant of boolean convolution can be implemented in time faster than the usual $O(n\log n)$ expected from FFT-based methods.