Existing approaches to distributed matrix computations involve allocating coded combinations of submatrices to worker nodes, to build resilience to stragglers and/or enhance privacy. In this study, we consider the challenge of preserving input sparsity in such approaches to retain the associated computational efficiency enhancements. First, we find a lower bound on the weight of coding, i.e., the number of submatrices to be combined to obtain coded submatrices to provide the resilience to the maximum possible number of stragglers (for given number of nodes and their storage constraints). Next we propose a distributed matrix computation scheme which meets this exact lower bound on the weight of the coding. Further, we develop controllable trade-off between worker computation time and the privacy constraint for sparse input matrices in settings where the worker nodes are honest but curious. Numerical experiments conducted in Amazon Web Services (AWS) validate our assertions regarding straggler mitigation and computation speed for sparse matrices.

翻译：现有分布式矩阵计算方法通过向工作节点分配子矩阵的编码组合，以构建容迟能力和/或增强隐私性。本研究聚焦于在此类方法中保持输入稀疏性以保留相关计算效率提升的挑战。首先，我们推导了编码权重的下界，即在给定节点数量及其存储约束下，为抵御最大可能数量的迟滞节点（即工作节点故障）而需组合的子矩阵数量。其次，我们提出一种满足该编码权重严格下界的分布式矩阵计算方案。进一步地，针对工作节点为"诚实但好奇"的场景，我们建立了稀疏输入矩阵的工作节点计算时间与隐私约束之间的可控权衡机制。在亚马逊云服务（AWS）上开展的数值实验验证了本方法在稀疏矩阵迟滞缓解与计算速度方面的有效性。