Let $G=(V,E)$ be a weighted undirected graph, with $n$ vertices. A distance oracle is a data structure that can quickly answer distance queries, with some stretch factor. A seminal work of \cite{TZ01}, given an integer $k\ge 1$, provides such an oracle with stretch $2k-1$, query time $O(k)$, and size $O(k\cdot n^{1+1/k})$. Furthermore, this oracle can also report a path in $G$ corresponding to the returned distance. In this paper we focus on vertex-labeled graphs, in which each vertex is given a label from a set $L$ of size $\ell$. A {\em vertex-label distance oracle} answers queries of the form $(v,λ)$, where $v\in V$ and $λ\in L$, by reporting (an approximation to) the distance from $v$ to the closest vertex of label $λ$. Following \cite{HLWY11}, it was shown in \cite{C12} that for any integer $k> 1$, there exists a vertex-label distance oracle with stretch $4k-5$, query time $O(k)$, and size $O(k\cdot n\cdot \ell^{1/k})$. This state-of-the-art result suffers from two main drawbacks: The stretch is roughly a factor of 2 larger than in \cite{TZ01}, and it is not path-reporting. We address these concerns in this work, and provide the following results: First, we devise a {\em path-reporting} vertex-label distance oracle, at the cost of a slight increase in stretch and size. For any constant $0<ε<1$, our oracle has stretch $(4k-5)\cdot(1+ε)$, query time $O(k)$, and size $O(n^{1+o(1)}\cdot \ell^{1/k})$. Second, we show how to improve the stretch to the optimal $2k-1$, at the cost of mildly increasing the query time. Specifically, we devise a vertex-label distance oracle with stretch $2k-1$, query time $O(\ell^{1/k}\cdot\log n)$, and size $O(k\cdot n\cdot \ell^{1/k})$. \end{itemize}

翻译：设 $G=(V,E)$ 为一个带权无向图，具有 $n$ 个顶点。距离预言机是一种能够快速回答距离查询（带有拉伸因子）的数据结构。文献 \cite{TZ01} 的开创性工作对给定整数 $k\ge 1$ 提供了拉伸为 $2k-1$、查询时间为 $O(k)$、大小为 $O(k\cdot n^{1+1/k})$ 的此类预言机。此外，该预言机还能报告 $G$ 中对应于返回距离的路径。本文聚焦于顶点标记图，其中每个顶点被赋予一个来自大小为 $\ell$ 的集合 $L$ 的标签。**顶点标签距离预言机** 回答形如 $(v,\lambda)$ 的查询（其中 $v\in V$ 且 $\lambda\in L$），报告从 $v$ 到最近标签为 $\lambda$ 的顶点的距离（或近似距离）。继 \cite{HLWY11} 之后，文献 \cite{C12} 证明：对任意整数 $k>1$，存在拉伸为 $4k-5$、查询时间为 $O(k)$、大小为 $O(k\cdot n\cdot \ell^{1/k})$ 的顶点标签距离预言机。这一最新成果存在两个主要缺陷：拉伸因子约为 \cite{TZ01} 的两倍，且不支持路径报告。本文着力解决这些问题，并取得以下成果：首先，我们设计了一种**路径报告**的顶点标签距离预言机，代价是拉伸因子和大小略有增加。对任意常数 $0<ε<1$，我们的预言机拉伸为 $(4k-5)\cdot(1+ε)$，查询时间为 $O(k)$，大小为 $O(n^{1+o(1)}\cdot \ell^{1/k})$。其次，我们展示了如何将拉伸因子优化至最优值 $2k-1$，代价是查询时间略有增加。具体而言，我们设计了一种拉伸为 $2k-1$、查询时间为 $O(\ell^{1/k}\cdot\log n)$、大小为 $O(k\cdot n\cdot \ell^{1/k})$ 的顶点标签距离预言机。\end{itemize}