arXiv 2106.122932021-06-23cs.DS

寻找单源最短$p$ -不相交路径：快速计算和稀疏搜索

Finding single-source shortest $p$-disjoint paths: fast computation and sparse preservers

Davide Bilò, Gianlorenzo D'Angelo, Luciano Gualà, Stefano Leucci, Guido Proietti, Mirko Rossi

备注：: 18 pages, 7 figures

摘要：

设$G$是一个有向图，它有$n$顶点，$m$边和非负边代价。给定$G$，一个固定的源顶点$s$，和一个正整数$p$，我们考虑对每个顶点$t\neq s$，计算$G$中从$s$到$t$的总代价最小的$p$边不交路的问题. Suurballe和Tarjan~[Networks，1984]通过设计还计算稀疏\emph{single-source $2$-multipath preserver}的$O(m+n\log n)$时间算法解决了$p=2$的上述问题，即，一个包含$2$边不相交路径的子图，该路径具有从$s$到$G$的每隔一个顶点的最小总成本。病例$p \geq 3$作为未决问题。我们研究了一般问题（$p\geq 2$），并证明了任何图允许一个稀疏的单源$p$ -多路径转换器与$p(n-1)$边。这个大小是最佳的，因为每个非根顶点$v$的入度必须至少为$p$。此外，我们设计了一个算法，需要$O(pn^2 (p + \log n))$的时间来计算$p$边缘不相交路径的最小总成本从源到所有其他顶点和最佳大小的单源$p$ -多路径转换器。我们的算法的运行时间优于一个自然的方法，解决$n-1$单对实例使用著名的\emph{successive shortest paths}算法的一个因素的$Θ(\frac{m}{np})$，如果$p=O(1)$和$m=Θ(n^2)$是渐近接近最优的。我们的结果自然扩展到$p$顶点不相交路径的情况下。

英文摘要：

Let $G$ be a directed graph with $n$ vertices, $m$ edges, and non-negative edge costs. Given $G$, a fixed source vertex $s$, and a positive integer $p$, we consider the problem of computing, for each vertex $t\neq s$, $p$ edge-disjoint paths of minimum total cost from $s$ to $t$ in $G$. Suurballe and Tarjan~[Networks, 1984] solved the above problem for $p=2$ by designing a $O(m+n\log n)$ time algorithm which also computes a sparse \emph{single-source $2$-multipath preserver}, i.e., a subgraph containing $2$ edge-disjoint paths of minimum total cost from $s$ to every other vertex of $G$. The case $p \geq 3$ was left as an open problem. We study the general problem ($p\geq 2$) and prove that any graph admits a sparse single-source $p$-multipath preserver with $p(n-1)$ edges. This size is optimal since the in-degree of each non-root vertex $v$ must be at least $p$. Moreover, we design an algorithm that requires $O(pn^2 (p + \log n))$ time to compute both $p$ edge-disjoint paths of minimum total cost from the source to all other vertices and an optimal-size single-source $p$-multipath preserver. The running time of our algorithm outperforms that of a natural approach that solves $n-1$ single-pair instances using the well-known \emph{successive shortest paths} algorithm by a factor of $Θ(\frac{m}{np})$ and is asymptotically near optimal if $p=O(1)$ and $m=Θ(n^2)$. Our results extend naturally to the case of $p$ vertex-disjoint paths.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文