We study dynamic algorithms for maintaining fundamental algebraic properties of matrices, specifically, rank, basis, and full-rank submatrices, with applications to maximum matching on dynamic graphs. Prior dynamic algorithms for rank achieve subquadratic update times but scale with the matrix dimension $n$, and could not always maintain the corresponding objects such as a basis or maximum full-rank submatrix. We present the first dynamic rank algorithms whose update time scales with the matrix rank $r$, achieving $\tilde O(r^{1.405})$ time per entry-update and $\tilde O(r^{1.528}+ z)$ per column-update, where $z$ is the number of changed entries. This extends to $\tilde O(|M|^{1.405})$ edge-update time to maintain the size $|M|$ of a maximum matching. We also give dynamic algorithms for maintaining a column-basis subject to column-updates and a maximum full-rank submatrix subject to entry-updates.

翻译：我们研究维护矩阵基本代数性质的动态算法，具体包括秩、基以及满秩子矩阵，并应用于动态图的最大匹配问题。已有的动态秩算法虽能达到次二次更新时间复杂度，但其复杂度与矩阵维度$n$相关，且无法始终维护对应的对象（如基或最大满秩子矩阵）。我们提出了首个更新时间复杂度与矩阵秩$r$相关的动态秩算法，在单元素更新中达到$\tilde O(r^{1.405})$时间，在列更新中达到$\tilde O(r^{1.528}+ z)$时间（其中$z$为发生变化的元素数量）。该成果可推广至边更新场景：维护最大匹配规模$|M|$的时间复杂度为$\tilde O(|M|^{1.405})$。我们还给出了在列更新场景下维护列基、在元素更新场景下维护最大满秩子矩阵的动态算法。