Given an $n$-vertex undirected graph $G=(V,E,w)$, and a parameter $k\geq1$, a path-reporting distance oracle (or PRDO) is a data structure of size $S(n,k)$, that given a query $(u,v)\in V^2$, returns an $f(k)$-approximate shortest $u-v$ path $P$ in $G$ within time $q(k)+O(|P|)$. Here $S(n,k)$, $f(k)$ and $q(k)$ are arbitrary functions. A landmark PRDO due to Thorup and Zwick, with an improvement of Wulff-Nilsen, has $S(n,k)=O(k\cdot n^{1+\frac{1}{k}})$, $f(k)=2k-1$ and $q(k)=O(\log k)$. The size of this oracle is $\Omega(n\log n)$ for all $k$. Elkin and Pettie and Neiman and Shabat devised much sparser PRDOs, but their stretch was polynomially larger than the optimal $2k-1$. On the other hand, for non-path-reporting distance oracles, Chechik devised a result with $S(n,k)=O(n^{1+\frac{1}{k}})$, $f(k)=2k-1$ and $q(k)=O(1)$. In this paper we make a dramatic progress in bridging the gap between path-reporting and non-path-reporting distance oracles. We devise a PRDO with size $S(n,k)=O(\lceil\frac{k\log\log n}{\log n}\rceil\cdot n^{1+\frac{1}{k}})$, stretch $f(k)=O(k)$ and query time $q(k)=O(\log\lceil\frac{k\log\log n}{\log n}\rceil)$. We can also have size $O(n^{1+\frac{1}{k}})$, stretch $O(k\cdot\lceil\frac{k\log\log n}{\log n}\rceil)$ and query time $q(k)=O(\log\lceil\frac{k\log\log n}{\log n}\rceil)$. Our results on PRDOs are based on novel constructions of approximate distance preservers, that we devise in this paper. Specifically, we show that for any $\epsilon>0$, any $k=1,2,...$, and any graph $G$ and a collection $\mathcal{P}$ of $p$ vertex pairs, there exists a $(1+\epsilon)$-approximate preserver with $O(\gamma(\epsilon,k)\cdot p+n\log k+n^{1+\frac{1}{k}})$ edges, where $\gamma(\epsilon,k)=(\frac{\log k}{\epsilon})^{O(\log k)}$. These new preservers are significantly sparser than the previous state-of-the-art approximate preservers due to Kogan and Parter.

翻译：给定一个 $n$ 个顶点的无向图 $G=(V,E,w)$ 和参数 $k\geq1$，路径报告型距离预言（PRDO）是一种规模为 $S(n,k)$ 的数据结构，对于查询 $(u,v)\in V^2$，能在时间 $q(k)+O(|P|)$ 内返回 $G$ 中一条 $f(k)$-近似最短 $u-v$ 路径 $P$。其中 $S(n,k)$、$f(k)$ 和 $q(k)$ 为任意函数。Thorup 与 Zwick 提出的里程碑式 PRDO（经 Wulff-Nilsen 改进）达到 $S(n,k)=O(k\cdot n^{1+\frac{1}{k}})$、$f(k)=2k-1$ 且 $q(k)=O(\log k)$。该预言对任意 $k$ 的规模均为 $\Omega(n\log n)$。Elkin 与 Pettie 以及 Neiman 与 Shabat 设计了更稀疏的 PRDO，但其拉伸因子多项式倍大于最优值 $2k-1$。另一方面，对于非路径报告型距离预言，Chechik 获得 $S(n,k)=O(n^{1+\frac{1}{k}})$、$f(k)=2k-1$ 且 $q(k)=O(1)$ 的结果。本文在弥合路径报告型与非路径报告型距离预言之间的差距上取得了重大进展。我们设计了一种 PRDO，其规模 $S(n,k)=O(\lceil\frac{k\log\log n}{\log n}\rceil\cdot n^{1+\frac{1}{k}})$、拉伸因子 $f(k)=O(k)$ 且查询时间 $q(k)=O(\log\lceil\frac{k\log\log n}{\log n}\rceil)$。我们还可实现规模 $O(n^{1+\frac{1}{k}})$、拉伸因子 $O(k\cdot\lceil\frac{k\log\log n}{\log n}\rceil)$ 且查询时间 $q(k)=O(\log\lceil\frac{k\log\log n}{\log n}\rceil)$。我们的 PRDO 成果基于本文提出的近似距离保持器的新型构造方法。具体而言，我们证明：对任意 $\epsilon>0$、任意 $k=1,2,\dots$、任意图 $G$ 以及包含 $p$ 个顶点对的集合 $\mathcal{P}$，存在一个具有 $O(\gamma(\epsilon,k)\cdot p+n\log k+n^{1+\frac{1}{k}})$ 条边的 $(1+\epsilon)$-近似保持器，其中 $\gamma(\epsilon,k)=(\frac{\log k}{\epsilon})^{O(\log k)}$。这些新型保持器比 Kogan 与 Parter 先前的最优近似保持器显著更稀疏。