Given an undirected possibly weighted $n$-vertex graph $G=(V,E)$ and a set $\mathcal{P}\subseteq V^2$ of pairs, a subgraph $S=(V,E')$ is called a ${\cal P}$-pairwise $\alpha$-spanner of $G$, if for every pair $(u,v)\in\mathcal{P}$ we have $d_S(u,v)\leq\alpha\cdot d_G(u,v)$. The parameter $\alpha$ is called the stretch of the spanner, and its size overhead is define as $\frac{|E'|}{|{\cal P}|}$. A surprising connection was recently discussed between the additive stretch of $(1+\epsilon,\beta)$-spanners, to the hopbound of $(1+\epsilon,\beta)$-hopsets. A long sequence of works showed that if the spanner/hopset has size $\approx n^{1+1/k}$ for some parameter $k\ge 1$, then $\beta\approx\left(\frac1\epsilon\right)^{\log k}$. In this paper we establish a new connection to the size overhead of pairwise spanners. In particular, we show that if $|{\cal P}|\approx n^{1+1/k}$, then a ${\cal P}$-pairwise $(1+\epsilon)$-spanner must have size at least $\beta\cdot |{\cal P}|$ with $\beta\approx\left(\frac1\epsilon\right)^{\log k}$ (a near matching upper bound was recently shown in \cite{ES23}). We also extend the connection between pairwise spanners and hopsets to the large stretch regime, by showing nearly matching upper and lower bounds for ${\cal P}$-pairwise $\alpha$-spanners. In particular, we show that if $|{\cal P}|\approx n^{1+1/k}$, then the size overhead is $\beta\approx\frac k\alpha$. A source-wise spanner is a special type of pairwise spanner, for which ${\cal P}=A\times V$ for some $A\subseteq V$. A prioritized spanner is given also a ranking of the vertices $V=(v_1,\dots,v_n)$, and is required to provide improved stretch for pairs containing higher ranked vertices. By using a sequence of reductions, we improve on the state-of-the-art results for source-wise and prioritized spanners.

翻译：给定一个无向可能带权重的$n$顶点图$G=(V,E)$和一个点对集合$\mathcal{P}\subseteq V^2$，子图$S=(V,E')$被称为$G$的$\mathcal{P}$-成对$\alpha$-稀疏子图，如果对每一对点$(u,v)\in\mathcal{P}$都有$d_S(u,v)\leq\alpha\cdot d_G(u,v)$。参数$\alpha$称为稀疏子图的伸缩比，其大小开销定义为$\frac{|E'|}{|{\cal P}|}$。近期发现$(1+\epsilon,\beta)$-稀疏子图的加性伸缩比与$(1+\epsilon,\beta)$-跳集(hopsets)的跳数之间存在令人惊讶的联系。一系列研究表明：若稀疏子图/跳集的大小约为$n^{1+1/k}$（其中参数$k\ge 1$），则$\beta\approx\left(\frac1\epsilon\right)^{\log k}$。本文建立了成对稀疏子图大小开销的新联系。具体而言，我们证明若$|{\cal P}|\approx n^{1+1/k}$，则$\mathcal{P}$-成对$(1+\epsilon)$-稀疏子图的大小至少为$\beta\cdot |{\cal P}|$，其中$\beta\approx\left(\frac1\epsilon\right)^{\log k}$（最近文献\cite{ES23}给出了近乎匹配的上界）。我们还将成对稀疏子图与跳集之间的联系推广到大伸缩比场景，通过给出$\mathcal{P}$-成对$\alpha$-稀疏子图的近乎匹配上下界。特别地，我们证明若$|{\cal P}|\approx n^{1+1/k}$，则大小开销为$\beta\approx\frac k\alpha$。源点稀疏子图是成对稀疏子图的一种特殊类型，其中$\mathcal{P}=A\times V$且$A\subseteq V$。优先级稀疏子图还额外给定顶点排序$V=(v_1,\dots,v_n)$，要求为包含更高排名顶点的点对提供更优的伸缩比。通过一系列归约，我们改进了源点稀疏子图和优先级稀疏子图的最新研究结果。