This paper considers a natural fault-tolerant shortest paths problem: for some constant integer $f$, given a directed weighted graph with no negative cycles and two fixed vertices $s$ and $t$, compute (either explicitly or implicitly) for every tuple of $f$ edges, the distance from $s$ to $t$ if these edges fail. We call this problem $f$-Fault Replacement Paths ($f$FRP). We first present an $\tilde{O}(n^3)$ time algorithm for $2$FRP in $n$-vertex directed graphs with arbitrary edge weights and no negative cycles. As $2$FRP is a generalization of the well-studied Replacement Paths problem (RP) that asks for the distances between $s$ and $t$ for any single edge failure, $2$FRP is at least as hard as RP. Since RP in graphs with arbitrary weights is equivalent in a fine-grained sense to All-Pairs Shortest Paths (APSP) [Vassilevska Williams and Williams FOCS'10, J.~ACM'18], $2$FRP is at least as hard as APSP, and thus a substantially subcubic time algorithm in the number of vertices for $2$FRP would be a breakthrough. Therefore, our algorithm in $\tilde{O}(n^3)$ time is conditionally nearly optimal. Our algorithm implies an $\tilde{O}(n^{f+1})$ time algorithm for the $f$FRP problem, giving the first improvement over the straightforward $O(n^{f+2})$ time algorithm. Then we focus on the restriction of $2$FRP to graphs with small integer weights bounded by $M$ in absolute values. Using fast rectangular matrix multiplication, we obtain a randomized algorithm that runs in $\tilde{O}(M^{2/3}n^{2.9153})$ time. This implies an improvement over our $\tilde{O}(n^{f+1})$ time arbitrary weight algorithm for all $f>1$. We also present a data structure variant of the algorithm that can trade off pre-processing and query time. In addition to the algebraic algorithms, we also give an $n^{8/3-o(1)}$ conditional lower bound for combinatorial $2$FRP algorithms in directed unweighted graphs.
翻译:本文认为存在一个自然的错误容忍性最短路径问题:对于某些固定整数 {formax}{formical $53{n3}美元时间算法,考虑到一个直接的加权图表,没有负周期,两个固定的螺旋美元和美元美元,计算(明确或隐含地)每张美元边缘图,从美元到美元,如果这些边缘失效,从美元到美元之间的距离。我们把这个问题称为美元与美元之间的替换路径(ffFR)(fR)(fR)(fR)(fRP) 。我们首先用美元(n) 平面图显示$2美元,没有负周期。由于$2FRP(美元) 直径直线的计算法程程程程程程程程程程程程程程程程程程程程程程程程程 。