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$G=(V,E,w)$，以及参数$k\geq1$，路径报告距离预言（PRDO）是一种大小为$S(n,k)$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$f(k)$近似的最短$u-v$路径$P$。这里$S(n,k)$、$f(k)$和$q(k)$是任意函数。Thorup和Zwick提出的里程碑式PRDO（经Wulff-Nilsen改进）具有$S(n,k)=O(k\cdot n^{1+\frac{1}{k}})$、$f(k)=2k-1$和$q(k)=O(\log k)$。该预言的大小对所有$k$均为$\Omega(n\log n)$。Elkin与Pettie以及Neiman与Shabat设计了稀疏得多的PRDO，但其拉伸倍数比最优值$2k-1$呈多项式倍数增大。另一方面，对于非路径报告距离预言，Chechik取得的结果具有$S(n,k)=O(n^{1+\frac{1}{k}})$、$f(k)=2k-1$和$q(k)=O(1)$。本文在弥合路径报告与非路径报告距离预言之间的差距方面取得了重大进展。我们设计了一种PRDO，其大小为$S(n,k)=O(\lceil\frac{k\log\log n}{\log n}\rceil\cdot n^{1+\frac{1}{k}})$，拉伸为$f(k)=O(k)$，查询时间为$q(k)=O(\log\lceil\frac{k\log\log n}{\log n}\rceil)$。我们还可实现大小为$O(n^{1+\frac{1}{k}})$，拉伸为$O(k\cdot\lceil\frac{k\log\log n}{\log n}\rceil)$，查询时间为$q(k)=O(\log\lceil\frac{k\log\log n}{\log n}\rceil)$。我们的PRDO成果基于本文提出的近似距离保持器的新颖构造。具体地，我们证明：对任意$\epsilon>0$、任意$k=1,2,\ldots$、任意图$G$以及包含$p$个顶点对的集合$\mathcal{P}$，存在一个$(1+\epsilon)$近似的保持器，其边数为$O(\gamma(\epsilon,k)\cdot p+n\log k+n^{1+\frac{1}{k}})$，其中$\gamma(\epsilon,k)=(\frac{\log k}{\epsilon})^{O(\log k)}$。这些新型保持器比Kogan和Parter先前最先进的近似保持器显著更稀疏。