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.
翻译:我们考虑的是Modular SubsetSum 问题: 给一个来自$$\mathb ⁇ m美元和目标整整数的多设 $x$, 决定是否存在一个以$(m\text{polog{m}m)美元计算的子X$子集。 红衣主教和Iacotono( SOSA'21)最近的独立工作为这一问题提供了简单和近线算法。 红衣主教和Iacono给出了一个随机化的算法, 以$( m\log m) 的时间运行, 而 Axiotis 等人( Axiotis etal) 给出了一个以$(m) (m) 美元运行的确定性算法算法算法, 以美元计算一个以$(m) 美元(m) 美元(m) 和 美元(molox) 运行的确定性算法算法。 我们提供两种结果的递解算法, 也就是以我们货币算算算算算算算的变数。