In this article, we provide a unified and simplified approach to derandomize central results in the area of fault-tolerant graph algorithms. Given a graph $G$, a vertex pair $(s,t) \in V(G)\times V(G)$, and a set of edge faults $F \subseteq E(G)$, a replacement path $P(s,t,F)$ is an $s$-$t$ shortest path in $G \setminus F$. For integer parameters $L,f$, a replacement path covering (RPC) is a collection of subgraphs of $G$, denoted by $\textit{G}_{L,f}=\{G_1,\ldots, G_r \}$, such that for every set $F$ of at most $f$ faults (i.e., $|F|\le f$) and every replacement path $P(s,t,F)$ of at most $L$ edges, there exists a subgraph $G_i\in \textit{G}_{L,f}$ that contains all the edges of $P$ and does not contain any of the edges of $F$. The covering value of the RPC $\textit{G}_{L,f}$ is then defined to be the number of subgraphs in $\textit{G}_{L,f}$. We present efficient deterministic constructions of $(L,f)$-RPCs whose covering values almost match the randomized ones, for a wide range of parameters. Our time and value bounds improve considerably over the previous construction of Parter (DISC 2019). We also provide an almost matching lower bound for the value of these coverings. A key application of our above deterministic constructions is the derandomization of the algebraic construction of the distance sensitivity oracle by Weimann and Yuster (FOCS 2010). The preprocessing and query time of the our deterministic algorithm nearly match the randomized bounds. This resolves the open problem of Alon, Chechik and Cohen (ICALP 2019).

翻译：本文提供了一种统一且简化的方法，用于对容错图算法领域的核心结果进行去随机化。给定图$G$、顶点对$(s,t) \in V(G)\times V(G)$以及边故障集合$F \subseteq E(G)$，替换路径$P(s,t,F)$是$G \setminus F$中从$s$到$t$的最短路径。对于整数参数$L,f$，替换路径覆盖(RPC)是$G$的子图集合，记为$\textit{G}_{L,f}=\{G_1,\ldots, G_r\}$，使得对于任意不超过$f$个故障的集合$F$（即$|F|\le f$）以及任意边数不超过$L$的替换路径$P(s,t,F)$，存在一个子图$G_i\in \textit{G}_{L,f}$包含$P$的所有边且不包含$F$的任何边。RPC $\textit{G}_{L,f}$的覆盖值定义为$\textit{G}_{L,f}$中的子图数量。对于广泛范围的参数，我们给出了$(L,f)$-RPC的有效确定性构造，其覆盖值几乎与随机化构造匹配。我们的时间复杂度和覆盖值界限相较于Parter (DISC 2019)的先前构造有显著提升。我们还为这些覆盖值提供了几乎匹配的下界。上述确定性构造的一个关键应用是Weimann与Yuster (FOCS 2010)的距离敏感预处理器代数构造的去随机化。我们确定性算法的预处理与查询时间几乎与随机化边界匹配。这解决了Alon、Chechik与Cohen (ICALP 2019)的开放问题。