We study applications of clustering (in particular the $k$-center clustering problem) in the design of efficient and practical deterministic algorithms for computing an approximate and the exact arithmetic matrix product of two 0-1 rectangular matrices $A$ and $B$ with clustered rows or columns, respectively. Let $\lambda_A$ and $\lambda_B$ denote the minimum maximum radius of a cluster in an $\ell$-center clustering of the rows of $A$ and in a $k$-center clustering of the columns of $B,$ respectively. In particular, assuming that the matrices have size $n\times n$, we obtain the following results. A simple deterministic algorithm that approximates each entry of the arithmetic matrix product of $A$ and $B$ within the additive error of at most $2\lambda_A$ in $O(n^2\ell)$ time or at most $2\lambda_B$ in $O(n^2k)$ time. A simple deterministic preprocessing of the matrices $A$ and $B$ in $O(n^2\ell)$ time or $O(n^2k)$ time such that a query asking for the exact value of an arbitrary entry of the arithmetic matrix product of $A$ and $B$ can be answered in $O(\lambda_A)$ time or $O(\lambda_B)$ time, respectively. A simple deterministic algorithm for the exact arithmetic matrix product of $A$ and $B$ running in time $O(n^2(\ell+k+\min\{\lambda_A,\lambda_B\}))$.

翻译：本文研究聚类（特别是$k$-中心聚类问题）在设计与实现高效、实用的确定性算法中的应用，用于计算两个0-1矩形矩阵$A$和$B$的近似及精确算术矩阵乘积，其中矩阵的行或列分别具有聚类结构。令$\lambda_A$和$\lambda_B$分别表示对$A$的行进行$\ell$-中心聚类以及对$B$的列进行$k$-中心聚类中，各聚类最小最大半径。特别地，假设矩阵尺寸为$n\times n$，我们得到以下结果：一个简单的确定性算法，可在$O(n^2\ell)$时间内以不超过$2\lambda_A$的加性误差近似$A$与$B$算术矩阵乘积的每个元素，或在$O(n^2k)$时间内以不超过$2\lambda_B$的加性误差进行近似；一个简单的确定性预处理方法，在$O(n^2\ell)$时间或$O(n^2k)$时间内处理矩阵$A$和$B$，使得针对$A$与$B$算术矩阵乘积任意元素的精确值查询，可分别在$O(\lambda_A)$时间或$O(\lambda_B)$时间内得到回答；一个简单的确定性算法，用于计算$A$与$B$的精确算术矩阵乘积，其运行时间为$O(n^2(\ell+k+\min\{\lambda_A,\lambda_B\}))$。