Given a constant number d of $n\times n$ matrices with total m non-infinity entries, we show how to construct in (essentially optimal) $\tilde{O}(mn+n^2)$ time a data structure that can compute in (essentially optimal) $\tilde{O}(n^2)$ time the distance product of these matrices after incrementing the value (possibly to infinity) of a constant number k of entries.

Our result is obtained by designing an oracle for single source replacement paths that is suited for short distances: Given a graph $G=(V,E)$, a source vertex s, and a shortest paths tree T of depth d rooted at s, the oracle can be constructed in $\tilde{O}(m{d}^k)$ time for any constant k. Then, given an arbitrary set S of at most $k=O(1)$ edges and an arbitrary vertex t, the oracle in $\tilde{O}(1)$ time either reports the length of the shortest s-to-t path in $(V,E\setminus S)$ or otherwise reports that any such shortest path must use more than d edges.

给定一个常数d的$ñ\times ñ$包含总共m个非无限项的矩阵，我们展示了如何构造（基本上是最优的）$\stackrel{〜}{Ø}（米ñ+{ñ}^2）$ 为可以计算的数据结构计时（基本上是最佳时间） $\stackrel{〜}{Ø}（{ñ}^2）$在增加常数k的条目的值（可能为无穷大）之后，对这些矩阵的距离乘积进行计时。

通过为单个数据源替换路径设计一个适用于短距离的预言片来获得我们的结果：给定一个图$G=（V，Ë）$，源顶点s和以s为根的深度为d的最短路径树T，可以在$\stackrel{〜}{Ø}（米d^{ķ}）$任何常数k的时间。然后，给定任意一组S ^最多的$ķ=Ø（\mathrm{1个}）$边和任意顶点t，$\stackrel{〜}{Ø}（\mathrm{1个}）$时间要么报告最短s- to- t路径的长度$（V，Ë\setminus \mathrm{小号}）$否则报告任何此类最短路径必须使用多于d条边。