稀疏非负卷积的简单拉斯维加斯算法
A Simple Las Vegas Algorithm for Sparse Nonnegative Convolution
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中文总结 AI 辅助
本文提出一种稀疏非负卷积的简单拉斯维加斯算法，基于稠密卷积、线性哈希及长度约简技术，其期望运行时间与现有算法匹配但结构更简单，尾边界更弱。
中文摘要 AI 辅助
设$A, B$为$\boldsymbol{Z}_{\boldsymbol{\u22650}}^n$中的非负向量，$t = |\boldsymbol{\text{supp}}(A \u2217 B)|$（$A \u2217 B$的支撑集大小）。本文提出一种拉斯维加斯算法，可在$O(t \boldsymbol{\text{log}} t)$期望时间内计算$A \u2217 B$；更一般地，对任意$0 < \boldsymbol{\u03b4} \boldsymbol{\u2264} \frac{1}{2}$，该算法在$O(t \boldsymbol{\text{log}} t \boldsymbol{\text{log}} \frac{1}{\boldsymbol{\u03b4}})$时间内终止的概率至少为$1 - \boldsymbol{\u03b4}$。算法使用稠密卷积、线性哈希及文献[BFN22]中的长度约简技术，核心是将索引表示为恒定维度$d$的无进位向量，其坐标大小为$O(t / \boldsymbol{\text{log}} t)$；取素数$p$大小为$\boldsymbol{\u03a9}(t / \boldsymbol{\text{log}} t)$时，哈希函数为该素数域$\boldsymbol{\text{F}}_p^d$中随机元素的内积，此哈希保持加法运算且碰撞概率恰好为$1/p$，通过向量矩相关恒等式可识别并恢复孤立项，与文献[BFN22]方法一致。本文算法的期望运行时间与Jin和Xu的文献[JX24]结果匹配，但使用的工具差异显著且算法更简单；需注意，文献[JX24]的算法终止于$O(t \boldsymbol{\text{log}} t)$时间的概率至少为$1 - \frac{1}{t}$，而本文算法的尾边界更弱。
英文摘要
Let $A, B \in \mathbb{Z}_{\ge 0}^n$ be nonnegative vectors and let $t = |\operatorname{supp}(A \star B)|$. We give a Las Vegas algorithm that computes $A \star B$ in $O(t \log t)$ expected time. More generally, for every $0 < δ\le \frac{1}{2}$, the algorithm terminates within $O(t \log t \log \frac{1}δ)$ time with probability at least $1 - δ$. The algorithm uses dense convolution, linear hashing, and the length reduction of \cite{BFN22}. Its main ingredient is a carry-free representation of the indices as vectors of constant dimension $d$ whose coordinates have size $O(t / \log t)$. We can then take our hash function to be the inner product with a random element of $\mathbb{F}_p^d$ for a prime $p$ of size $Ω(t / \log t)$: this preserves addition and gives collision probability exactly $1/p$, while identities regarding the moments of the vectors identify and recover the isolated terms as in \cite{BFN22}. Our expected running time matches that of Jin and Xu~\cite{JX24} while using substantially different tools and yielding a simpler algorithm. Note that their algorithm also terminates within $O(t \log t)$ time with probability at least $1 - \frac{1}{t}$, while our tail bound is weaker.
