We initiate a study of solving a row/column diagonally dominant (RDD/CDD) linear system $Mx=b$ in sublinear time, with the goal of estimating $t^{\top}x^*$ for a given vector $t\in R^n$ and a specific solution $x^*$. This setting naturally generalizes the study of sublinear-time solvers for symmetric diagonally dominant (SDD) systems [AKP19] to the asymmetric case. Our first contributions are characterizations of the problem's mathematical structure. We express a solution $x^*$ via a Neumann series, prove its convergence, and upper bound the truncation error on this series through a novel quantity of $M$, termed the maximum $p$-norm gap. This quantity generalizes the spectral gap of symmetric matrices and captures how the structure of $M$ governs the problem's computational difficulty. For systems with bounded maximum $p$-norm gap, we develop a collection of algorithmic results for locally approximating $t^{\top}x^*$ under various scenarios and error measures. We derive these results by adapting the techniques of random-walk sampling, local push, and their bidirectional combination, which have proved powerful for special cases of solving RDD/CDD systems, particularly estimating PageRank and effective resistance on graphs. Our general framework yields deeper insights, extended results, and improved complexity bounds for these problems. Notably, our perspective provides a unified understanding of Forward Push and Backward Push, two fundamental approaches for estimating random-walk probabilities on graphs. Our framework also inherits the hardness results for sublinear-time SDD solvers and local PageRank computation, establishing lower bounds on the maximum $p$-norm gap or the accuracy parameter. We hope that our work opens the door for further study into sublinear solvers, local graph algorithms, and directed spectral graph theory.

翻译：我们首次研究了在亚线性时间内求解行/列对角占优线性系统 $Mx=b$ 的问题，目标是在给定向量 $t\in R^n$ 和特定解 $x^*$ 的情况下估计 $t^{\top}x^*$。这一设定自然地将对称对角占优系统亚线性时间求解器[AKP19]的研究推广到非对称情形。我们的首要贡献在于对该问题数学结构的刻画。我们通过诺伊曼级数表示解 $x^*$，证明其收敛性，并通过引入 $M$ 的一个新度量——最大 $p$-范数间隙，对该级数的截断误差给出上界。该度量推广了对称矩阵的谱间隙概念，并揭示了 $M$ 的结构如何控制问题的计算难度。对于具有有界最大 $p$-范数间隙的系统，我们针对不同场景和误差度量，开发了一系列局部逼近 $t^{\top}x^*$ 的算法结果。这些结果是通过改进随机游走采样、局部推送及其双向组合技术而得到的，这些技术已在求解RDD/CDD系统的特殊案例（特别是估计图的PageRank和有效电阻）中展现出强大效能。我们的通用框架为这些问题提供了更深刻的见解、扩展性结果和改进的复杂度边界。值得注意的是，我们的视角为前向推送与后向推送这两种估计图上随机游走概率的基本方法提供了统一理解。该框架还继承了亚线性时间SDD求解器和局部PageRank计算的硬度结果，建立了关于最大 $p$-范数间隙或精度参数的下界。我们希望本研究能为亚线性求解器、局部图算法及有向谱图论的进一步研究开启新的方向。