A spanner of a graph is a subgraph that preserves lengths of shortest paths up to a multiplicative distortion. For every $k$, a spanner with size $O(n^{1+1/k})$ and stretch $(2k+1)$ can be constructed by a simple centralized greedy algorithm, and this is tight assuming Erd\H{o}s girth conjecture. In this paper we study the problem of constructing spanners in a local manner, specifically in the Local Computation Model proposed by Rubinfeld et al. (ICS 2011). We provide a randomized Local Computation Agorithm (LCA) for constructing $(2r-1)$-spanners with $\tilde{O}(n^{1+1/r})$ edges and probe complexity of $\tilde{O}(n^{1-1/r})$ for $r \in \{2,3\}$, where $n$ denotes the number of vertices in the input graph. Up to polylogarithmic factors, in both cases, the stretch factor is optimal (for the respective number of edges). In addition, our probe complexity for $r=2$, i.e., for constructing a $3$-spanner, is optimal up to polylogarithmic factors. Our result improves over the probe complexity of Parter et al. (ITCS 2019) that is $\tilde{O}(n^{1-1/2r})$ for $r \in \{2,3\}$. Both our algorithms and the algorithms of Parter et al. use a combination of neighbor-probes and pair-probes in the above-mentioned LCAs. For general $k\geq 1$, we provide an LCA for constructing $O(k^2)$-spanners with $\tilde{O}(n^{1+1/k})$ edges using $O(n^{2/3}\Delta^2)$ neighbor-probes, improving over the $\tilde{O}(n^{2/3}\Delta^4)$ algorithm of Parter et al. By developing a new randomized LCA for graph decomposition, we further improve the probe complexity of the latter task to be $O(n^{2/3-(1.5-\alpha)/k}\Delta^2)$, for any constant $\alpha>0$. This latter LCA may be of independent interest.

翻译：图的一个稀疏子图是保持最短路径长度至多乘性畸变的子图。对于每个$k$，通过简单的集中式贪心算法可以构造一个规模为$O(n^{1+1/k})$、伸缩因子为$(2k+1)$的稀疏子图，且该结果在假设 Erdős 周长猜想下是紧的。本文研究在局部计算模型（Rubinfeld 等人提出，ICS 2011）中局部构造稀疏子图的问题。我们提出一种随机化局部计算算法（LCA），用于构造$(2r-1)$-稀疏子图，其边数为$\tilde{O}(n^{1+1/r})$，探测复杂度为$\tilde{O}(n^{1-1/r})$，其中$r \in \{2,3\}$，$n$表示输入图的顶点数。忽略多对数因子，两种情况下伸缩因子均为最优（对应各自边数）。此外，对于$r=2$（即构造$3$-稀疏子图），我们的探测复杂度在多对数因子意义下也是最优的。该结果改进了 Parter 等人（ITCS 2019）针对$r \in \{2,3\}$的$\tilde{O}(n^{1-1/2r})$探测复杂度。我们的算法与 Parter 等人的算法均结合使用了上述 LCA 中的邻域探测和对探测。对于一般的$k\geq 1$，我们提供一种 LCA 用于构造$O(k^2)$-稀疏子图，其边数为$\tilde{O}(n^{1+1/k})$，使用$O(n^{2/3}\Delta^2)$次邻域探测，改进了 Parter 等人$\tilde{O}(n^{2/3}\Delta^4)$的算法。通过开发一种新的图分解随机化 LCA，我们进一步将后者的探测复杂度改进为$O(n^{2/3-(1.5-\alpha)/k}\Delta^2)$，其中$\alpha>0$为任意常数。该 LCA 可能具有独立的研究价值。