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

Abstract

The modular subset sum problem consists of deciding, given a modulus $m$, a multiset $S$ of $n$ integers in $\mathrm{0..}m-1$, and a target integer $t$, whether there exists a subset of $S$ with elements summing to $t\mathrm{mod}m$, 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 Erdős-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).

Revised version of the original which appeared at the 2021 SIAM Symposium on Simplicity in Algorithms (SOSA21)