Lower Bounds on Sparse Spanners, Emulators, and Diameter-Reducing Shortcuts
Shang-En Huang, Seth Pettie · SIAM Journal on Discrete Mathematics · 2021
We prove better lower bounds on additive spanners and emulators, which are lossy compression schemes for undirected graphs, as well as lower bounds on shortcut sets, which reduce the diameter of directed graphs. We prove that any $O(n)$-size shortcut set cannot bring the diameter below $\Omega(n^{1/6})$ and that any $O(m)$-size shortcut set cannot bring it below $\Omega(n^{1/11})$. These improve Hesse's [ Proceedings of the 14 th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), Baltimore, MD, 2003] lower bound of $\Omega(n^{1/17})$. By combining these constructions with Abboud and Bodwin's [ J. ACM, 64 (2017), 28] edge-splitting technique, we get additive stretch lower bounds of $+\Omega(n^{1/11})$ for $O(n)$-size spanners and $+\Omega(n^{1/18})$ for $O(n)$-size emulators. These improve Abboud and Bodwin's $+\Omega(n^{1/22})$ lower bounds for both spanners and emulators.