Given a weighted graph $G$, a $(\beta,\varepsilon)$-hopset $H$ is an edge set such that for any $s,t \in V(G)$, where $s$ can reach $t$ in $G$, there is a path from $s$ to $t$ in $G \cup H$ which uses at most $\beta$ hops whose length is in the range $[dist_G(s,t), (1+\varepsilon)dist_G(s,t)]$. We break away from the traditional question that asks for a hopset that achieves small $|H|$ and instead study its sensitivity, a new quality measure which, informally, is the maximum number of times a vertex (or edge) is bypassed by an edge in $H$. The highlights of our results are: (i) $(\widetilde{O}(\sqrt{n}),0)$-hopsets on undirected graphs with $O(\log n)$ sensitivity, complemented with a lower bound showing that $\widetilde{O}(\sqrt{n})$ is tight up to polylogarithmic factors for any construction with polylogarithmic sensitivity. (ii) $(n^{o(1)},\varepsilon)$-hopsets on undirected graphs with $n^{o(1)}$ sensitivity for any $\varepsilon > 0$ that is at least inverse polylogarithmic, complemented with a lower bound on the tradeoff between $\beta, \varepsilon$, and the sensitivity. (iii) $\widetilde{O}(\sqrt{n})$-shortcut sets on directed graphs with $O(\log n)$ sensitivity, complemented with a lower bound showing that $\beta = \widetilde{\Omega}(n^{1/3})$ for any construction with polylogarithmic sensitivity. We believe hopset sensitivity is a natural measure in and of itself, and could potentially find use in a diverse range of contexts. More concretely, the notion of hopset sensitivity is also directly motivated by the Differentially Private All Sets Range Queries problem. Our result for $O(\log n)$ sensitivity $(\widetilde{O}(\sqrt{n}),0)$-hopsets on undirected graphs immediately improves the current best-known upper bound on utility from $\widetilde{O}(n^{1/3})$ to $\widetilde{O}(n^{1/4})$ in the pure-DP setting, which is tight up to polylogarithmic factors.

翻译：给定一个加权图$G$，一个$(\beta,\varepsilon)$-跳集$H$是一个边集，使得对于任意$s,t \in V(G)$（其中$s$在$G$中可达$t$），在$G \cup H$中存在一条从$s$到$t$的路径，该路径最多使用$\beta$跳，且其长度在区间$[dist_G(s,t), (1+\varepsilon)dist_G(s,t)]$内。我们摒弃了传统上追求跳集规模$|H|$最小化的问题，转而研究其敏感度——一种新的质量度量，非正式地说，它是指一个顶点（或边）被$H$中的边绕过的最大次数。我们研究结果的核心亮点包括：（i）在无向图上构造具有$O(\log n)$敏感度的$(\widetilde{O}(\sqrt{n}),0)$-跳集，并辅以下界证明：对于任何具有多对数敏感度的构造，$\widetilde{O}(\sqrt{n})$的跳数在除去多对数因子后是紧的。（ii）对于任何至少为逆多对数的$\varepsilon > 0$，在无向图上构造具有$n^{o(1)}$敏感度的$(n^{o(1)},\varepsilon)$-跳集，并辅以关于$\beta$、$\varepsilon$与敏感度之间权衡的下界。（iii）在有向图上构造具有$O(\log n)$敏感度的$\widetilde{O}(\sqrt{n})$-捷径集，并辅以下界证明：对于任何具有多对数敏感度的构造，$\beta = \widetilde{\Omega}(n^{1/3})$。我们相信跳集敏感度本身是一个自然的度量标准，并有可能在多种不同情境中找到应用。更具体地说，跳集敏感度的概念也直接受到差分隐私所有集合范围查询问题的启发。我们在无向图上关于$O(\log n)$敏感度$(\widetilde{O}(\sqrt{n}),0)$-跳集的结果，立即将纯差分隐私设置下当前已知的最佳效用上界从$\widetilde{O}(n^{1/3})$改进到$\widetilde{O}(n^{1/4})$，该结果在除去多对数因子后是紧的。