arXiv 2002.068122020-02cs.DS

多顶点失效的近似距离预言机

Approximate Distance Oracles Subject to Multiple Vertex Failures

Ran Duan, Yong Gu, Hanlin Ren

备注：: SODA'21

摘要：

给定$n$顶点和$m$边的无向图$G=(V,E)$，在$[1,W]$中，我们构造顶点敏感距离预言机（VSDO），这是预处理图的数据结构，并回答以下类型的查询：给定源顶点$u$、目标顶点$v$和一批$d$失败顶点$D$，输出$G-D$中$u$和$v$之间的距离（近似值）（即，去除$D$中顶点的图$G$）。如果Oracle始终保持$δ_{G-D}(u,v)\le\tildeδ(u,v)\leα\cdotδ_{G-D}(u,v)$，则它具有拉伸$α$，其中$δ_{G-D}(u,v)$是$G-D$中$u$和$v$之间的实际距离，而$\tildeδ(u,v)$是Oracle报告的距离。在本文中，我们构造有效的VSDO的任何数量的$d$的故障。对于任何常数$c\geq 1$，我们提出了两个预言：$\bullet$第一个预言的大小$n^{2+1/c}(\log n/ε)^{O(d)}\cdot \log W$，回答查询在${\rm poly}(\log n,d^c,\log\log W,ε^{-1})$时间，并有拉伸$1+ε$，任何常数$ε>0$。$\bullet$第二个oracle的大小为$n^{2+1/c}{\rm poly}(\log (nW),d)$，在${\rm poly}(\log n,d^c,\log\log W)$时间内回答查询，并具有扩展${\rm poly}(\log n,d)$。这两个预言机都可以在其空间复杂度的时间多项式中进行预处理。这些结果是一般无向图中任意常数个顶点失效的多对数查询时间的第一个近似距离预言。以前有$(1+ε)$ -近似$d$ -边缘敏感距离预言机[Escherhik et al. 2017]在$d$边缘失败时回答距离查询，其具有大小$O(n^2(\log n/ε)^d\cdot d\log W)$和查询时间${\rm poly}(\log n, d, \log\log W)$。

英文摘要：

Given an undirected graph $G=(V,E)$ of $n$ vertices and $m$ edges with weights in $[1,W]$, we construct vertex sensitive distance oracles (VSDO), which are data structures that preprocess the graph, and answer the following kind of queries: Given a source vertex $u$, a target vertex $v$, and a batch of $d$ failed vertices $D$, output (an approximation of) the distance between $u$ and $v$ in $G-D$ (that is, the graph $G$ with vertices in $D$ removed). An oracle has stretch $α$ if it always holds that $δ_{G-D}(u,v)\le\tildeδ(u,v)\leα\cdotδ_{G-D}(u,v)$, where $δ_{G-D}(u,v)$ is the actual distance between $u$ and $v$ in $G-D$, and $\tildeδ(u,v)$ is the distance reported by the oracle. In this paper we construct efficient VSDOs for any number $d$ of failures. For any constant $c\geq 1$, we propose two oracles: $\bullet$ The first oracle has size $n^{2+1/c}(\log n/ε)^{O(d)}\cdot \log W$, answers a query in ${\rm poly}(\log n,d^c,\log\log W,ε^{-1})$ time, and has stretch $1+ε$, for any constant $ε>0$. $\bullet$ The second oracle has size $n^{2+1/c}{\rm poly}(\log (nW),d)$, answers a query in ${\rm poly}(\log n,d^c,\log\log W)$ time, and has stretch ${\rm poly}(\log n,d)$. Both of these oracles can be preprocessed in time polynomial in their space complexity. These results are the first approximate distance oracles of poly-logarithmic query time for any constant number of vertex failures in general undirected graphs. Previously there are $(1+ε)$-approximate $d$-edge sensitive distance oracles [Chechik et al. 2017] answering distance queries when $d$ edges fail, which have size $O(n^2(\log n/ε)^d\cdot d\log W)$ and query time ${\rm poly}(\log n, d, \log\log W)$.

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