具有对数拉伸和大小为O(n loglog n)的路径报告距离预言机
Path-Reporting Distance Oracles with Logarithmic Stretch and Size O(n loglog n)
AI总结：
本文提出了一种新型路径报告距离预言机（PRDO），其大小和拉伸显著逼近非路径报告距离预言机的最优界限，并基于新型近似距离保持器构造，大幅改进了现有数据结构的稀疏性。
摘要：
给定一个$n$个顶点的无向图$G=(V,E,w)$和一个参数$k\geq1$，路径报告距离预言机（或PRDO）是一个大小为$S(n,k)$的数据结构，给定查询$(u,v)\in V^2$，在$q(k)+O(|P|)$时间内返回$G$中一条$f(k)$-近似最短$u-v$路径$P$。此处$S(n,k)$、$f(k)$和$q(k)$是任意函数。由Thorup和Zwick提出、经Wulff-Nilsen改进的里程碑式PRDO具有$S(n,k)=O(k\cdot n^{1+\frac{1}{k}})$、$f(k)=2k-1$和$q(k)=O(\log k)$。对所有$k$，该预言机大小为$\Omega(n\log n)$。Elkin和Pettie以及Neiman和Shabat设计了更稀疏的PRDO，但其拉伸比最优的$2k-1$大出多项式级别。另一方面，对于非路径报告距离预言机，Chechik设计了一个$S(n,k)=O(n^{1+\frac{1}{k}})$、$f(k)=2k-1$且$q(k)=O(1)$的结果。本文在弥合路径报告与非路径报告距离预言机之间的差距方面取得了巨大进展。我们设计了一个大小为$S(n,k)=O(\lceil\frac{k\log\log n}{\log n}\rceil\cdot n^{1+\frac{1}{k}})$、拉伸$f(k)=O(k)$且查询时间$q(k)=O(\log\lceil\frac{k\log\log n}{\log n}\rceil)$的PRDO。我们也可以实现大小$O(n^{1+\frac{1}{k}})$、拉伸$O(k\cdot\lceil\frac{k\log\log n}{\log n}\rceil)$和查询时间$q(k)=O(\log\lceil\frac{k\log\log n}{\log n}\rceil)$。我们关于PRDO的结果基于本文设计的新型近似距离保持器构造。具体而言，我们证明对于任意$\epsilon>0$、任意$k=1,2,...$、任意图$G$和包含$p$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$\gamma(\epsilon,k)=(\frac{\log k}{\epsilon})^{O(\log k)}$。这些新保持器比Kogan和Parter先前的最先进近似保持器稀疏得多。
英文摘要：
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 $Ω(n\log n)$ 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 $ε>0$, any $k=1,2,...$, and any graph $G$ and a collection $\mathcal{P}$ of $p$ vertex pairs, there exists a $(1+ε)$-approximate preserver with $O(γ(ε,k)\cdot p+n\log k+n^{1+\frac{1}{k}})$ edges, where $γ(ε,k)=(\frac{\log k}ε)^{O(\log k)}$. These new preservers are significantly sparser than the previous state-of-the-art approximate preservers due to Kogan and Parter.
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。
