超稀疏近加性仿真器
Ultra-Sparse Near-Additive Emulators
摘要：
近加性（又名$(1+ε,β)$ -）仿真器和spectron是一种基本的图形算法构造，具有许多用于计算近似最短路径的应用程序以及分布式，流式和动态设置中的相关问题。近加性仿真器的已知构造使得人们能够在它们的稀疏性（即，边数）和附加拉伸$β$。具体地，对于任何一对参数$ε>0$、$ κ=1,2,\dots$，可以具有具有$O(n^{1+1/κ})$边的$(1+ε,β)$ -仿真器，具有$β= \left(\frac{\log κ}ε\right)^{\log κ}$。在它们的sparterly，这些仿真器使用$c\cdot n$边缘，对于一些恒定的$c\geq 2$。我们收紧这一界限，并表明，事实上正是$n^{1+1/κ}$边缘足够。特别地，我们的仿真器可以是\emph{ultra-sparse}，即，我们可以有一个模拟器与$n+o(n)$边缘和$β= \left(\frac{\log {\log n}}{ε}\right)^{\log {\log n}(1+o(1))}$。我们还在CONGEST模型中设计了一个分布式确定性算法，该算法在低多项式时间内构建这些仿真器（即，在$O(n^ρ)$时间中，对于任意小常数参数$ρ>0$）。最后，我们还改进了PODC'19论文[ElkinM 19]中设计的$(1+ε,β)$ -spectrometry的最先进的分布式确定性\congest -模型构造。具体地，[ElkinM 19]的spectrum具有$O(β\cdot n^{1+1/κ})$边缘，即，在它们的间隔处，它们使用$ O\left(\frac{\log {\log n}}{ε}\right)^{\log {\log n}}\cdot n$边缘。在本文中，我们设计了一个有效的分布式确定性CONGEST模型算法，建立这样的空间与$O(n^{1+1/κ})$边缘$κ= O\left(\frac{\log n}{\log ^{(3)}n}\right)$。在它们的稀疏度上，这些spatial仅使用$O(n\cdot {\log {\log n}})$边缘。
英文摘要：
Near-additive (aka $(1+ε,β)$-) emulators and spanners are a fundamental graph-algorithmic construct, with numerous applications for computing approximate shortest paths and related problems in distributed, streaming and dynamic settings. Known constructions of near-additive emulators enable one to trade between their sparsity (i.e., number of edges) and the additive stretch $β$. Specifically, for any pair of parameters $ε>0$, $ κ=1,2,\dots$, one can have a $(1+ε,β)$-emulator with $O(n^{1+1/κ})$ edges, with $β= \left(\frac{\log κ}ε\right)^{\log κ}$. At their sparsest, these emulators employ $c\cdot n$ edges, for some constant $c\geq 2$. We tighten this bound, and show that in fact precisely $n^{1+1/κ}$ edges suffice. In particular, our emulators can be \emph{ultra-sparse}, i.e., we can have an emulator with $n+o(n)$ edges and $β= \left(\frac{\log {\log n}}{ε}\right)^{\log {\log n}(1+o(1))}$. We also devise a distributed deterministic algorithm in the CONGEST model that builds these emulators in low polynomial time (i.e., in $O(n^ρ)$ time, for an arbitrarily small constant parameter $ρ>0$). Finally, we also improve the state-of-the-art distributed deterministic \congest-model construction of $(1+ε,β)$-spanners devised in the PODC'19 paper [ElkinM19]. Specifically, the spanners of [ElkinM19] have $O(β\cdot n^{1+1/κ})$ edges, i.e., at their sparsest they employ $ O\left(\frac{\log {\log n}}{ε}\right)^{\log {\log n}}\cdot n$ edges. In this paper, we devise an efficient distributed deterministic CONGEST-model algorithm that builds such spanners with $O(n^{1+1/κ})$ edges for $κ= O\left(\frac{\log n}{\log ^{(3)}n}\right)$. At their sparsest, these spanners employ only $O(n\cdot {\log {\log n}})$ edges.
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。
