AI 中文总结
该研究针对高维ℓ_p空间几何生成树的拉伸-大小权衡，给出已有2跳生成树下界的简单证明并扩展至带Steiner顶点的情形，还建立了有界跳与通用生成树的关联，推导了无跳限制生成树的下界结论。
AI 中文摘要
我们研究了高维ℓ_p空间中几何生成树的拉伸-大小权衡关系。我们的主要贡献是给出了Har-Peled、Indyk和Sidiropoulos[SODA 2013]提出的一个下界的简单证明：在ℓ_2范数下，点集{0,1}^d的每个2跳t-生成树至少具有(2^d)^{1+Ω(1/t²)}条边。我们的证明进一步将该结果扩展到带有Steiner顶点的生成树。此外，我们建立了有界跳生成树与通用生成树之间的联系，具体如下：如果一个n点度量空间的每个子集Y都有一个边数至多为μ|Y|的t-生成树，那么该度量空间存在一个大小为O(n(μ+log n))的O(t)跳O(t)-生成树。由此可得，一个度量空间的跳受限生成树下界蕴含着其某个子集的无跳限制生成树下界。
英文摘要
We study the stretch--size tradeoff for geometric spanners in high-dimensional $\ell_p$ spaces. Our main contribution is a simple proof of a lower bound shown by Har-Peled, Indyk, and Sidiropoulos [SODA 2013]: Every $2$-hop $t$-spanner of the pointset $\{0,1\}^d$ under $\ell_2$ norm has at least $(2^d)^{1+Ω(1/t^2)}$ edges. Our proof further extends this result to spanners with Steiner vertices. In addition, we establish a connection between bounded-hop spanners and general spanners, as follows. If every subset $Y$ of an $n$-point metric has a $t$-spanner with at most $μ|Y|$ edges, then the metric has an $O(t)$-hop $O(t)$-spanner of size $O(n(μ+\log n))$. Consequently, hop-restricted spanner lower bounds for a metric imply lower bounds without hop restriction for one of its subsets.
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