Modular Subset Sum, Dynamic Strings, and Zero-Sum Sets

TL;DR

This paper presents a faster, simpler randomized algorithm for the modular subset sum problem, improving previous time bounds and applying it to find zero-sum subsets in Ramsey theory efficiently.

Contribution

It introduces a new $O(m \,\log m)$-time algorithm for modular subset sum, utilizing a simplified data structure called the Data Dependent Tree.

Findings

Achieves $O(m \log m)$ runtime with high probability

Provides a faster algorithm for zero-sum subset detection

Simplifies the dynamic strings approach with a new data structure

Abstract

The modular subset sum problem consists of deciding, given a modulus $m$, a multiset $S$ of $n$ integers in $0..m-1$, and a target integer $t$, whether there exists a subset of $S$ with elements summing to $t \mod m $, and to report such a set if it exists. We give a simple $O(m \log m)$-time with high probability (w.h.p.) algorithm for the modular subset sum problem. This builds on and improves on a previous $O(m \log^7 m)$ w.h.p. algorithm from Axiotis, Backurs, Jin, Tzamos, and Wu (SODA 19). Our method utilizes the ADT of the dynamic strings structure of Gawrychowski et al. (SODA~18). However, as this structure is rather complicated we present a much simpler alternative which we call the Data Dependent Tree. As an application, we consider the computational version of a fundamental theorem in zero-sum Ramsey theory. The Erd\H{o}s-Ginzburg-Ziv Theorem states that a multiset of $2n - 1$ integers always contains a subset of cardinality exactly $n$ whose values sum to a multiple of $n$. We give an algorithm for finding such a subset in time $O(n \log n)$ w.h.p. which improves on an $O(n^2)$ algorithm due to Del Lungo, Marini, and Mori (Disc. Math. 09).