Low Sensitivity Hopsets
<|reference_start|>Low Sensitivity Hopsets: 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.<|reference_end|>
@article{ashvinkumar2024low,
    title={Low Sensitivity Hopsets},
    author={Vikrant Ashvinkumar, Aaron Bernstein, Chengyuan Deng, Jie Gao, Nicole
  Wein},
    journal={arXiv preprint arXiv:2407.10249},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.10249},
    primaryClass={cs.DS}
}