Finding a Solution to the Erd\H{o}s-Ginzburg-Ziv Theorem in Linear Time

TL;DR

This paper presents a new deterministic linear-time algorithm for finding a specific subsequence in integer sequences, improving upon previous algorithms with higher time complexities.

Contribution

It introduces a linear-time algorithm for a prime subset-sum problem, enabling a linear-time solution to the Erd ext{o}s-Ginzburg-Ziv theorem.

Findings

Achieves linear-time complexity for the problem.

Uses a novel arithmetic-progression representation of sums.

Extends the prime algorithm to arbitrary moduli.

Abstract

The Erd\H{o}s-Ginzburg-Ziv theorem states that every sequence of 2n - 1 integers contains a subsequence of length n whose sum is divisible by n. Choi, Kang, and Lim gave a simple deterministic O(n log n) algorithm for finding such a subsequence, and Leung recently improved this to O(n log log log n). We give a deterministic linear-time algorithm. The core is a linear-time algorithm for the following prime target subset-sum problem: given p - 1 nonzero residues in Z_p and a target residue, find a subset with the prescribed sum. Our algorithm maintains a compact arithmetic-progression representation of reachable sums. When two progressions intersect, a bounded Frobenius interval in their sum allows them to be merged into one longer progression, with enough growth to pay for the update. When the representation either contains a full progression or covers all nonzero residues, the target residue is recovered constructively. The standard multiplicative reduction then extends the prime algorithm to arbitrary moduli.