We study rectangular matrix multiplication in the low-bandwidth model of distributed computing. There are $n$ computers; initially the input matrices are distributed evenly between computers, and in each communication round every computer can send and receive an $O(\log n)$-bit message. Eventually each computer must output its designated part of the product matrix. While prior work has focused primarily on square $n \times n$ multiplication under various sparsity assumptions, we study rectangular instances with no sparsity assumption. We denote by $\langle a,b,c\rangle$ the task of multiplying an $a\times b$ matrix by a $b\times c$ matrix in this model. We concentrate on two natural aspect ratios, $\langle n,d,n\rangle$ and $\langle d,n,d\rangle$, for $d \le n$, and we study how the round complexity depends on $n$ and $d$. When $d \to n$, both $\langle n,d,n\rangle$ and $\langle d,n,d\rangle$ approach $\langle n,n,n\rangle$, which is the usual task of multiplying square matrices. If we consider multiplication over semirings, the current best upper bound in that case is $O(n^{4/3})$ rounds, and there is a trivial unconditional lower bound of $Ω(n)$. We show that for $\langle d,n,d\rangle$, we can achieve the complexity of $\tilde O(d^{4/3})$, which seems like a natural generalization of the upper bound $\tilde O(n^{4/3})$ when $d=n$. However, the case of $\langle n,d,n\rangle$ is fundamentally different, and also exhibits a phase transition. We show that for $d \le \sqrt{n}$, the complexity of $\langle n,d,n\rangle$ is $Θ(d \sqrt{n})$; we have matching upper and lower bounds. However, the behavior is genuinely different in the region $d \ge \sqrt{n}$, where the upper bound is $O(d^{2/3} n^{2/3})$.

翻译：我们研究分布式计算低带宽模型中的矩形矩阵乘法问题。设有$n$台计算机：初始时输入矩阵均匀分布在各计算机上，每轮通信中每台计算机可发送和接收一条$O(\log n)$比特的消息。最终每台计算机需输出其指定部分的乘积矩阵。现有工作主要关注在不同稀疏性假设下的方阵$n \times n$乘法，而本文研究无稀疏性假设的矩形实例。我们用$\langle a,b,c\rangle$表示在该模型下将$a\times b$矩阵与$b\times c$矩阵相乘的任务。我们聚焦于两种自然的宽高比：$\langle n,d,n\rangle$和$\langle d,n,d\rangle$，其中$d \le n$，并研究轮复杂度如何依赖于$n$和$d$。当$d \to n$时，$\langle n,d,n\rangle$和$\langle d,n,d\rangle$均趋近于$\langle n,n,n\rangle$——即常规的方阵乘法任务。若考虑半环上的乘法，该情形下当前最优上界为$O(n^{4/3})$轮，且存在平凡的$\Omega(n)$无条件下界。我们证明对于$\langle d,n,d\rangle$，可实现$\tilde O(d^{4/3})$的复杂度，这看似是对$d=n$时上界$\tilde O(n^{4/3})$的自然推广。然而$\langle n,d,n\rangle$情形则存在本质差异，并表现出相变特性。我们证明：当$d \le \sqrt{n}$时，$\langle n,d,n\rangle$的复杂度为$\Theta(d \sqrt{n})$，且上下界匹配。但在$d \ge \sqrt{n}$区域，行为截然不同，此时上界为$O(d^{2/3} n^{2/3})$。