Faster Deterministic Modular Subset Sum

TL;DR

This paper introduces a specialized data structure called the shift-tree to improve algorithms for the Modular Subset Sum problem, achieving faster deterministic solutions with near-linear runtime.

Contribution

The paper presents the shift-tree data structure and applies it to develop faster deterministic algorithms for Modular Subset Sum, matching previous randomized algorithms' efficiency.

Findings

Deterministic algorithm runs in O(m log m * α(m)) time.

Randomized algorithm matches O(m log m) runtime.

Shift-tree simplifies handling text problems in subset sum algorithms.

Abstract

We consider the Modular Subset Sum problem: given a multiset $X$ of integers from $\mathbb{Z}_m$ and a target integer $t$, decide if there exists a subset of $X$ with a sum equal to $t \pmod{m}$. Recent independent works by Cardinal and Iacono (SOSA'21), and Axiotis et al. (SOSA'21) provided simple and near-linear algorithms for this problem. Cardinal and Iacono gave a randomized algorithm that runs in $O(m \log m)$ time, while Axiotis et al. gave a deterministic algorithm that runs in $O(m \text{ polylog } m)$ time. Both results work by reduction to a text problem, which is solved using a dynamic strings data structure. In this work, we develop a simple data structure, designed specifically to handle the text problem that arises in the algorithms for Modular Subset Sum. Our data structure, which we call the shift-tree, is a simple variant of a segment tree. We provide both a hashing-based and a deterministic variant of the shift-trees. We then apply our data structure to the Modular Subset Sum problem and obtain two algorithms. The first algorithm is Monte-Carlo randomized and matches the $O(m \log m)$ runtime of the Las-Vegas algorithm by Cardinal and Iacono. The second algorithm is fully deterministic and runs in $O(m \log m \cdot \alpha(m))$ time, where $\alpha$ is the inverse Ackermann function.