Algebraic techniques have had an important impact on graph algorithms so far. Porting them, e.g., the matrix inverse, into the dynamic regime improved best-known bounds for various dynamic graph problems. In this paper, we develop new algorithms for another cornerstone algebraic primitive, the Frobenius normal form (FNF). We apply our developments to dynamic and fault-tolerant exact distance oracle problems on directed graphs. For generic matrices $A$ over a finite field accompanied by an FNF, we show (1) an efficient data structure for querying submatrices of the first $k\geq 1$ powers of $A$, and (2) a near-optimal algorithm updating the FNF explicitly under rank-1 updates. By representing an unweighted digraph using a generic matrix over a sufficiently large field (obtained by random sampling) and leveraging the developed FNF toolbox, we obtain: (a) a conditionally optimal distance sensitivity oracle (DSO) in the case of single-edge or single-vertex failures, providing a partial answer to the open question of Gu and Ren [ICALP'21], (b) a multiple-failures DSO improving upon the state of the art (vd. Brand and Saranurak [FOCS'19]) wrt. both preprocessing and query time, (c) improved dynamic distance oracles in the case of single-edge updates, and (d) a dynamic distance oracle supporting vertex updates, i.e., changing all edges incident to a single vertex, in $\tilde{O}(n^2)$ worst-case time and distance queries in $\tilde{O}(n)$ time.

翻译：代数技术迄今对图算法产生了重要影响。将其（例如矩阵求逆）移植到动态场景中，改进了多种动态图问题的最优已知界。本文针对另一基石性代数原语——弗罗贝尼乌斯标准形（FNF），开发了新算法。我们将这些进展应用于有向图上的动态及容错精确距离预言机问题。对于有限域上配备FNF的泛化矩阵$A$，我们展示了：（1）一种高效数据结构，用于查询$A$前$k\geq 1$次幂的子矩阵；（2）一种近乎最优的算法，可在秩为1的更新下显式更新FNF。通过使用泛化矩阵（随机采样获得）在足够大的域上表示无权重有向图，并利用所开发的FNF工具包，我们获得：（a）单边或单顶点失效情形下的条件最优距离灵敏度预言机（DSO），部分回答了Gu与Ren的开放问题[ICALP'21]；（b）一种多重失效DSO，在预处理和查询时间方面均改进了现有技术（参见Brand与Saranurak [FOCS'19]）；（c）单边更新情形下改进的动态距离预言机；（d）一种支持顶点更新的动态距离预言机，即改变单个顶点关联的所有边，最坏情况下更新时间为$\tilde{O}(n^2)$，距离查询时间为$\tilde{O}(n)$。