极小直径欧氏稀疏生成子的最优时间构造
An Optimal-Time Construction of Euclidean Sparse Spanners with Tiny Diameter
摘要：
在STOC'95 \cite{ADMSS95}中，Arya等人示出了对于$\mathbb R^d$中的任何$n$点集合，直径至多为2（分别为3）的$(1+ε)$ -点和$O(n \log n)$边（分别为，$O(n \log \log n)$边缘）可以在$O(n \log n)$时间内构建。此外，在\cite{ADMSS95,NS07}中显示，对于任何$k \ge 4$，可以在$O(n (\log n) 2^k α_k(n))$时间内构建直径至多为$2k$和$O(n 2^k α_k(n))$边缘的$(1+ε)$ -链。函数$α_k$是在原始递归层次的$\lfloor k/2 \rfloor$级的某个函数的逆，其中$α_0(n) = \lceil n/2 \rceil, α_1(n) = \left\lceil \sqrt{n} \right\rceil, α_2(n) = \lceil \log{n} \rceil, α_3(n) = \lceil \log\log{n} \rceil, α_4(n) = \log^* n$、\ldots等。还已知\cite{NS07}，如果允许二次时间，则可以改进这些界限。具体地，对于任何$k \ge 4$，可以在$O(n^2)$时间\cite{NS07}中构造直径至多为$k$和$O(n k α_k(n))$边的$(1+ε)$ -矩阵。在这一领域的一个主要的开放问题是，是否可以在时间内构造$O(n \log n + n k α_k(n))$的$(1+ε)$ -最多$k$和$O(n k α_k(n))$边缘的直径。在本文中，我们肯定地回答这个问题。此外，事实上，我们提供了一个更强的结果。具体来说，我们表明，对于任何$k \ge 4$，一个$(1+ε)$ -最$k$和$O(n α_k(n))$边缘的直径，可以建立在最佳时间$O(n \log n)$。在整个参数范围内，我们的spectrum的直径和边数之间的权衡是紧到常数因子的。
英文摘要：
In STOC'95 \cite{ADMSS95} Arya et al.\ showed that for any set of $n$ points in $\mathbb R^d$, a $(1+ε)$-spanner with diameter at most 2 (respectively, 3) and $O(n \log n)$ edges (resp., $O(n \log \log n)$ edges) can be built in $O(n \log n)$ time. Moreover, it was shown in \cite{ADMSS95,NS07} that for any $k \ge 4$, one can build in $O(n (\log n) 2^k α_k(n))$ time a $(1+ε)$-spanner with diameter at most $2k$ and $O(n 2^k α_k(n))$ edges. The function $α_k$ is the inverse of a certain function at the $\lfloor k/2 \rfloor$th level of the primitive recursive hierarchy, where $α_0(n) = \lceil n/2 \rceil, α_1(n) = \left\lceil \sqrt{n} \right\rceil, α_2(n) = \lceil \log{n} \rceil, α_3(n) = \lceil \log\log{n} \rceil, α_4(n) = \log^* n$, \ldots, etc. It is also known \cite{NS07} that if one allows quadratic time then these bounds can be improved. Specifically, for any $k \ge 4$, a $(1+ε)$-spanner with diameter at most $k$ and $O(n k α_k(n))$ edges can be constructed in $O(n^2)$ time \cite{NS07}. A major open problem in this area is whether one can construct within time $O(n \log n + n k α_k(n))$ a $(1+ε)$-spanner with diameter at most $k$ and $O(n k α_k(n))$ edges. In this paper we answer this question in the affirmative. Moreover, in fact, we provide a stronger result. Specifically, we show that for any $k \ge 4$, a $(1+ε)$-spanner with diameter at most $k$ and $O(n α_k(n))$ edges can be built in optimal time $O(n \log n)$. The tradeoff between the diameter and number of edges of our spanners is tight up to constant factors in the entire range of parameters.
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