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arXiv 2108.00102cs.DS
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一般图在（近似）线性时间上的近似最优生成子

Near-Optimal Spanners for General Graphs in (Nearly) Linear Time

Hung Le, Shay Solomon

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37 pages, 5 figures
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摘要：

设$G = (V,E,w)$是$|V| = n$顶点和$|E| = m$边上的加权无向图，$k \ge 1$是任意整数，$ε< 1$是任意参数.我们提出了以下结果的快速建设的spectrometry与接近最佳的稀疏性和亮度，最终在这方面的一长串工作。(By近似最优我们指的是在Erdos '围长猜想下的最优，并且忽略$ε$依赖性。- 存在（确定性的）算法用于构造具有接近最优稀疏度$O(n^{1/k} \log(1/ε)/ε))$的$G$的$(2k-1)(1+ε)$ -空间。第一个算法可以在时间$O(mα(m,n) \log(1/ε)/ε) + SORT(m))$内在指针机器模型中实现，其中$α( , )$是两参数逆阿克曼函数，$SORT(m)$是对$m$整数进行排序所需的时间。第二种算法可以在时间$O(m \log(1/ε)/ε))$内在WORD RAM模型中实现。- 有一种（确定性的）算法用于为$G$构建$(2k-1)(1+ε)$ -矩阵，它在稀疏度和亮度上都达到了$O(n^{1/k}\mathrm{poly}(1/ε))$的接近最优的界限.该算法可在时间$O(mα(m,n) \mathrm{poly}(1/ε) + SORT(m))$内的指针机模型和时间$O(m α(m,n) \mathrm{poly}(1/ε))$内的WORD RAM模型中实现。以前最快的$(2k-1)(1+ε)$ -spectrum的构造具有接近最优的稀疏性，导致运行时间为$O(\min\{m(n^{1+1/k}) + n\log n,k n^{2+1/k}\})$，即使与亮度无关。重要的是，拉伸$2k-1$的贪婪遍历具有稀疏$O(n^{1/k})$ --不依赖于$ε$，但其运行时是$O(m(n^{1+1/k} + n\log n))$。此外，任何$(2k-1)$-STATE的最新亮度边界都很差，即使不考虑稀疏性和运行时间。

英文摘要：

Let $G = (V,E,w)$ be a weighted undirected graph on $|V| = n$ vertices and $|E| = m$ edges, let $k \ge 1$ be any integer, and let $ε< 1$ be any parameter. We present the following results on fast constructions of spanners with near-optimal sparsity and lightness, which culminate a long line of work in this area. (By near-optimal we mean optimal under Erdős' girth conjecture and disregarding the $ε$-dependencies.) - There are (deterministic) algorithms for constructing $(2k-1)(1+ε)$-spanners for $G$ with a near-optimal sparsity of $O(n^{1/k} \log(1/ε)/ε))$. The first algorithm can be implemented in the pointer-machine model within time $O(mα(m,n) \log(1/ε)/ε) + SORT(m))$, where $α( , )$ is the two-parameter inverse-Ackermann function and $SORT(m)$ is the time needed to sort $m$ integers. The second algorithm can be implemented in the WORD RAM model within time $O(m \log(1/ε)/ε))$. - There is a (deterministic) algorithm for constructing a $(2k-1)(1+ε)$-spanner for $G$ that achieves a near-optimal bound of $O(n^{1/k}\mathrm{poly}(1/ε))$ on both sparsity and lightness. This algorithm can be implemented in the pointer-machine model within time $O(mα(m,n) \mathrm{poly}(1/ε) + SORT(m))$ and in the WORD RAM model within time $O(m α(m,n) \mathrm{poly}(1/ε))$. The previous fastest constructions of $(2k-1)(1+ε)$-spanners with near-optimal sparsity incur a runtime of is $O(\min\{m(n^{1+1/k}) + n\log n,k n^{2+1/k}\})$, even regardless of the lightness. Importantly, the greedy spanner for stretch $2k-1$ has sparsity $O(n^{1/k})$ -- with no $ε$-dependence whatsoever, but its runtime is $O(m(n^{1+1/k} + n\log n))$. Moreover, the state-of-the-art lightness bound of any $(2k-1)$-spanner is poor, even regardless of the sparsity and runtime.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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