顶点标记图的路径报告距离 oracle
Path-Reporting Distance Oracles for Vertex-Labeled Graphs
AI总结：
本文针对顶点标记图提出路径报告距离 oracle，改进了 stretch 和查询时间，实现了最优 stretch 2k-1，同时保持了高效的存储空间。
摘要：
设 G=(V,E) 为一个加权无向图，有 n 个顶点。距离 oracle 是一种数据结构，可以快速回答距离查询，具有一定的 stretch 因子。文献 [TZ01] 给出了一种 oracle，具有 stretch 2k-1，查询时间 O(k)，大小 O(k·n^{1+1/k})。此外，这种 oracle 还可以报告 G 中对应返回距离的路径。本文专注于顶点标记图，其中每个顶点被赋予一个来自集合 L 的标签。顶点标签距离 oracle 回答 (v,λ) 的查询，其中 v∈V 且 λ∈L，通过报告（近似）v 到 λ 最近顶点的距离。根据 [HLWY11]，文献 [C12] 显示，对于任何整数 k>1，存在一种顶点标签距离 oracle，具有 stretch 4k-5，查询时间 O(k)，大小 O(k·n·ℓ^{1/k})。这一最新结果有两个主要缺点：stretch 约比 [TZ01] 大一倍，且不报告路径。本文解决了这些问题，给出了以下结果：首先，我们设计了一种路径报告的顶点标签距离 oracle，以略微增加 stretch 和大小为代价。对于任何常数 0<ε<1，我们的 oracle 有 stretch (4k-5)·(1+ε)，查询时间 O(k)，大小 O(n^{1+o(1)}·ℓ^{1/k})。其次，我们展示了如何将 stretch 改进到最优 2k-1，以略微增加查询时间为代价。具体来说，我们设计了一种顶点标签距离 oracle，具有 stretch 2k-1，查询时间 O(ℓ^{1/k}·log n)，大小 O(k·n·ℓ^{1/k})。
英文摘要：
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}
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。
