Better Lower Bounds for Shortcut Sets and Additive Spanners via an Improved Alternation Product
Better Lower Bounds for Shortcut Sets and Additive Spanners via an Improved Alternation Product
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通过改进的交替乘积为快捷方式集和附加扳手提供更好的下界
DOI：
10.1137/1.9781611977073.131
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发表时间：
2021
期刊：
影响因子：
--
通讯作者：
Zixuan Xu
中科院分区：
文献类型：
--
作者：
Kevin Lu;V. V. Williams;Nicole Wein;Zixuan Xu
我们获得了改进的下限添加剂speries，添加剂仿真器，直径减少快捷集。生成器和仿真器是稀疏图，它们近似地保持给定图的距离。快捷集是一组边，当将其添加到有向图时，会减小其直径。Huang和Pettie [SWAT 2018]给出了这三种结构的先前最知名的下界。对于$O（n）$大小的空间，我们将加法拉伸的下界从$\Omega（n^{1/11}）$改进为$\Omega（n^{2/21}）$。对于$O（n）$大小的仿真器，我们将加法拉伸的下限从$\Omega（n^{1/18}）$提高到$\Omega（n^{1/16}）$。对于$O（m）$大小的捷径集，我们将图直径的下界从$\Omega（n^{1/11}）$提高到$\Omega（n^{1/8}）$。我们的关键技术贡献，这是我们所有的边界的基础上，是一个图形产品称为交替产品的改进。
We obtain improved lower bounds for additive spanners, additive emulators, and diameter-reducing shortcut sets. Spanners and emulators are sparse graphs that approximately preserve the distances of a given graph. A shortcut set is a set of edges that when added to a directed graph, decreases its diameter. The previous best known lower bounds for these three structures are given by Huang and Pettie [SWAT 2018]. For $O(n)$-sized spanners, we improve the lower bound on the additive stretch from $\Omega(n^{1/11})$ to $\Omega(n^{2/21})$. For $O(n)$-sized emulators, we improve the lower bound on the additive stretch from $\Omega(n^{1/18})$ to $\Omega(n^{1/16})$. For $O(m)$-sized shortcut sets, we improve the lower bound on the graph diameter from $\Omega(n^{1/11})$ to $\Omega(n^{1/8})$. Our key technical contribution, which is the basis of all of our bounds, is an improvement of a graph product known as an alternation product.
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