Polynomial coding has been proposed as a solution to the straggler mitigation problem in distributed matrix multiplication. Previous works in the literature employ univariate polynomials to encode matrix partitions. Such schemes greatly improve the speed of distributed computing systems by making the task completion time to depend only on the fastest workers. However, the work done by the slowest workers, which fails to finish the task assigned to them, is completely ignored. In order to exploit the partial computations of the slower workers, we further decompose the overall matrix multiplication task into even smaller subtasks to better fit workers' storage and computation capacities. In this work, we show that univariate schemes fail to make an efficient use of the storage capacity and we propose bivariate polynomial codes. We show that bivariate polynomial codes are a more natural choice to accommodate the additional decomposition of subtasks, as well as, heterogeneous storage and computation resources at workers. However, in contrast to univariate polynomial decoding, for multivariate interpolation guarantying decodability is much harder. We propose two bivartiate polynomial schemes. The first scheme exploits the fact that bivariate interpolation is always possible for rectangular grid of points. We obtain the rectangular grid of points at the cost of allowing some redundant computations. For the second scheme, we relax the decoding constraint, and require decodability for almost all choices of evaluation points. We present interpolation sets satisfying the almost decodability conditions for certain storage configurations of workers. Our numerical results show that bivariate polynomial coding considerably reduces the completion time of distributed matrix multiplication.

中文翻译：

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异构编码计算系统中利用落后者的双变量多项式编码

多项式编码已被提议作为分布式矩阵乘法中落后者缓解问题的解决方案。以前的文献工作使用单变量多项式来编码矩阵分区。这种方案通过使任务完成时间仅依赖于最快的工人，大大提高了分布式计算系统的速度。然而，最慢的工人所做的工作，没有完成分配给他们的任务，就被完全忽略了。为了利用较慢的工作人员的部分计算，我们进一步将整个矩阵乘法任务分解为更小的子任务，以更好地适应工作人员的存储和计算能力。在这项工作中，我们表明单变量方案无法有效利用存储容量，我们提出了双变量多项式代码。我们表明，二元多项式代码是更自然的选择，以适应子任务的额外分解，以及工作人员的异构存储和计算资源。然而，与单变量多项式解码相比，多变量插值保证可解码性要困难得多。我们提出了两个二元多项式方案。第一种方案利用了这样一个事实，即对于点的矩形网格总是可以进行双变量插值。我们以允许一些冗余计算为代价获得点的矩形网格。对于第二种方案，我们放宽了解码约束，并要求几乎所有评估点选择的可解码性。我们提出了满足工人某些存储配置的几乎可解码条件的插值集。