Near-Additive Spanners and Near-Exact Hopsets, A Unified View
Near-Additive Spanners and Near-Exact Hopsets, A Unified View
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近加法扳手和近精确 Hopsets，统一视图
DOI：
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发表时间：
2020
期刊：
影响因子：
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通讯作者：
Ofer Neiman
中科院分区：
文献类型：
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作者：
Michael Elkin;Ofer Neiman
给定一个{\ em未加权}无向图$ g =（v，e）$和一对参数$ \ epsilon> 0 $，$ \ beta = 1,2，\ ldots $，一个子graph $ g'=（ v，h）$，$ h \ subseteq e $，$ g $是{\ em $（1+ \ epsilon，\ beta）$ - SPANNER}（aka，a {\ em近近加速启动器}）$ g $，如果每个$ u，v \ in v $，$ $ d_ {g'} （u，v）\ le（1+ \ epsilon）d_g（u，v） + \ beta〜。$$。 \ cite {ep01}对于上述任何$ n $ vertex $ g $，以及任何$ \ epsilon> 0 $ and $ \ kappa = 1,2，\ ldots $，存在$（1+ \ epsilon，， \ beta）$ - SPANNER $ G'$ with $ o _ {\ epsilon，\ kappa}（n^{1+1/\ kappa}）$ edges，带有$$ \ beta = \ beta = \ beta_ {ep} = \ lesg } \ right）^{\ log \ kappa -2}〜。$$此键保留最新的及其对$ \ epsilon $（对于小$ \ kappa $的情况）的依赖性在\ cite {abp18}中被证明很紧。
给定一个{\ em加权}无方向的图$ g =（v，e，\ omega）$和一对参数$ \ epsilon> 0 $，$ \ beta = 1,2，\ ldots $，图$ g '=（v，h，\ omega'）$是{\ em $（1+ \ epsilon，\ beta）$ - hopset} （aka，a {\ em接近ex-exact hopset}）$ g $，如果每个$ u，v \ in v $，$$ d_g（u，v）\ le d_ {g \ cup g'}^{（ \ beta）}（u，v）\ le（1+ \ epsilon）d_g（u，v）〜，$$，$ d_ {g \ cup g'}^{（\ beta）}（u，v）$代表$ \ beta $ - （hop）在Union Graph $ g \ cup cup g'$中$ u $和$ v $之间的结合距离。它在\ cite {en16}中显示，对于任何$ n $ vertex $ g $和$ \ epsilon $和$ \ kappa $，如上所述，存在$（1+ \ epsilon，\ beta）$ - 与Hopset一起使用$ \ tilde {o}（n^{1+1/\ kappa}）$ edge，$ \ beta = \ beta_ {ep} $。
\ cite {ep01}和\ cite {en16}的两个结果不仅非常相似，而且它们的证明技术也是如此。此外，Thorup-Zwick后来在\ cite {en19，hp17}中显示了近加性跨度\ cite {tz06}的构造，以提供类似的杂种（\ cite {tz06}）特性。
在这项调查中，我们探讨了这种有趣的现象，绘制用于这些结果的基本证明技术，并突出介绍开放的问题。
Given an {\em unweighted} undirected graph $G = (V,E)$, and a pair of parameters $\epsilon > 0$, $\beta = 1,2,\ldots$, a subgraph $G' =(V,H)$, $H \subseteq E$, of $G$ is a {\em $(1+\epsilon,\beta)$-spanner} (aka, a {\em near-additive spanner}) of $G$ if for every $u,v \in V$, $$d_{G'}(u,v) \le (1+\epsilon)d_G(u,v) + \beta~.$$ It was shown in \cite{EP01} that for any $n$-vertex $G$ as above, and any $\epsilon > 0$ and $\kappa = 1,2,\ldots$, there exists a $(1+\epsilon,\beta)$-spanner $G'$ with $O_{\epsilon,\kappa}(n^{1+1/\kappa})$ edges, with $$\beta = \beta_{EP} = \left({{\log \kappa} \over \epsilon}\right)^{\log \kappa - 2}~.$$ This bound remains state-of-the-art, and its dependence on $\epsilon$ (for the case of small $\kappa$) was shown to be tight in \cite{ABP18}.
Given a {\em weighted} undirected graph $G = (V,E,\omega)$, and a pair of parameters $\epsilon > 0$, $\beta = 1,2,\ldots$, a graph $G'= (V,H,\omega')$ is a {\em $(1+\epsilon,\beta)$-hopset} (aka, a {\em near-exact hopset}) of $G$ if for every $u,v \in V$, $$d_G(u,v) \le d_{G\cup G'}^{(\beta)}(u,v) \le (1+\epsilon)d_G(u,v)~,$$ where $ d_{G\cup G'}^{(\beta)}(u,v)$ stands for a $\beta$-(hop)-bounded distance between $u$ and $v$ in the union graph $G \cup G'$. It was shown in \cite{EN16} that for any $n$-vertex $G$ and $\epsilon$ and $\kappa$ as above, there exists a $(1+\epsilon,\beta)$-hopset with $\tilde{O}(n^{1+1/\kappa})$ edges, with $\beta = \beta_{EP}$.
Not only the two results of \cite{EP01} and \cite{EN16} are strikingly similar, but so are also their proof techniques. Moreover, Thorup-Zwick's later construction of near-additive spanners \cite{TZ06} was also shown in \cite{EN19,HP17} to provide hopsets with analogous (to that of \cite{TZ06}) properties.
In this survey we explore this intriguing phenomenon, sketch the basic proof techniques used for these results, and highlight open questions.
次线性加法扳手下界的层次结构
DOI：
10.1137/1.9781611974782.36
发表时间：
2017
期刊：
SODA 2017
影响因子：
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作者：
Abboud, Amir;Bodwin, Greg;Pettie, Seth
通讯作者：
Pettie, Seth