Fast and Simple Modular Subset Sum

TL;DR

This paper introduces two simple, efficient algorithms for the Modular Subset Sum problem, achieving near-linear time complexity using elementary data structures, and also applies these techniques to improve algorithms for the All Pairs Non-Decreasing Paths problem.

Contribution

The paper presents the first simple near-linear time algorithms for Modular Subset Sum, one randomized and one deterministic, using elementary data structures and techniques.

Findings

Randomized algorithm runs in O(m log^2 m) time.

Deterministic algorithm runs in O(m polylog m) time.

Techniques extend to algorithms for the All Pairs Non-Decreasing Paths problem.

Abstract

We revisit the Subset Sum problem over the finite cyclic group $\mathbb{Z}_m$ for some given integer $m$. A series of recent works has provided near-optimal algorithms for this problem under the Strong Exponential Time Hypothesis. Koiliaris and Xu (SODA'17, TALG'19) gave a deterministic algorithm running in time $\tilde{O}(m^{5/4})$, which was later improved to $O(m \log^7 m)$ randomized time by Axiotis et al. (SODA'19). In this work, we present two simple algorithms for the Modular Subset Sum problem running in near-linear time in $m$, both efficiently implementing Bellman's iteration over $\mathbb{Z}_m$. The first one is a randomized algorithm running in time $O(m \log^2 m)$, that is based solely on rolling hash and an elementary data-structure for prefix sums; to illustrate its simplicity we provide a short and efficient implementation of the algorithm in Python. Our second solution is a deterministic algorithm running in time $O(m\ \mathrm{polylog}\ m)$, that uses dynamic data structures for string manipulation. We further show that the techniques developed in this work can also lead to simple algorithms for the All Pairs Non-Decreasing Paths Problem (APNP) on undirected graphs, matching the near-optimal running time of $\tilde{O}(n^2)$ provided in the recent work of Duan et al. (ICALP'19).