AI 中文总结
研究在双倍维度度量空间中计算最小生成树(1+ε)近似问题，给出确定性算法，改进了运行时间对ε的依赖，还改进了有界维欧几里得度量的最佳运行时间，且指出有界双倍度量中MST最大度特性。
AI 中文摘要
最小生成树（MST）问题是度量空间和图上最基本的优化问题之一。我们研究在双倍维度为ddim的n点度量空间(X,d)中计算MST的(1+ε)近似问题。在双倍度量中，先前的确定性算法运行时间依赖于ε^(-O(ddim))。我们给出一种确定性算法，能在2^(O(ddim))n(log n + ε^(-1)log^4(1/ε))时间内计算MST的(1+ε)近似。对于有界双倍维度，这改进了先前对ε的依赖，从ε^(-O(ddim))到本质上与ε^(-1)线性相关。作为特殊情况，我们的结果改进了Arya和Mount在SODA'16中给出的有界维欧几里得度量的先前最佳确定性运行时间，几乎快了ε^(-1)倍。我们还表明，与有界维欧几里得空间不同，有界双倍度量中的MST可以有任意大的最大度，而每个双倍度量都允许有最大度为2^(O(ddim))log(1/ε)的(1+ε)近似MST。
英文摘要
The minimum spanning tree (MST) problem is one of the most basic optimization problems on metric spaces and graphs. We study the problem of computing a $(1+ε)$-approximation to the MST of an $n$-point metric space $(X, \mathbf{d})$ of doubling dimension $\mathrm{ddim}$. In doubling metrics, previous deterministic algorithms incur a running time with dependence $ε^{-O(\mathrm{ddim})}$. We give a deterministic algorithm that computes a $(1+ε)$-approximation to MST in time $2^{O(\mathrm{ddim})} n \bigl(\log n + ε^{-1} \log^4(1/ε)\bigr)$. For bounded doubling dimension, this improves the previous dependence on $ε$ from $ε^{-O(\mathrm{ddim})}$ to essentially linear in $ε^{-1}$. Moreover, as a special case, our result improves the previous best deterministic running time for bounded-dimensional Euclidean metrics due to Arya and Mount~[SODA'16] by almost a factor of $ε^{-1}$. We also show that, unlike in bounded-dimensional Euclidean spaces, MSTs in bounded doubling metrics can have arbitrarily large maximum degree, while every doubling metric nevertheless admits a $(1+ε)$-approximate MST of maximum degree $2^{O(\mathrm{ddim})}\log(1/ε)$.
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