We study \emph{local computation algorithms (LCAs)} for constructing spanning trees. In this setting, the goal is to locally determine, for each edge $ e \in E $, whether it belongs to a spanning tree $ T $ of the input graph $ G $, where $ T $ is defined implicitly by $ G $ and the randomness of the algorithm. It is known that LCAs for spanning trees do not exist in general graphs, even for simple graph families. We identify a natural and well-studied class of graphs -- \emph{expander graphs} -- that do admit \emph{sublinear-time} LCAs for spanning trees. This is perhaps surprising, as previous work on expanders only succeeded in designing LCAs for \emph{sparse spanning subgraphs}, rather than full spanning trees. We design an LCA with probe complexity $ O\left(\sqrt{n}\left(\frac{\log^2 n}{φ^2} + d\right)\right)$ for graphs with conductance at least $ φ$ and maximum degree at most $ d $ (not necessarily constant), which is nearly optimal when $φ$ and $d$ are constants, since $Ω(\sqrt{n})$ probes are necessary even for expanders. Next, we show that for the natural class of \emph{\ER graphs} $ G(n, p) $ with $ np = n^δ $ for any constant $ δ> 0 $ (which are expanders with high probability), the $ \sqrt{n} $ lower bound can be bypassed. Specifically, we give an \emph{average-case} LCA for such graphs with probe complexity $ \tilde{O}(\sqrt{n^{1 - δ}})$. Finally, we extend our techniques to design LCAs for the \emph{minimum spanning tree (MST)} problem on weighted expander graphs. Specifically, given a $d$-regular unweighted graph $\bar{G}$ with sufficiently strong expansion, we consider the weighted graph $G$ obtained by assigning to each edge an independent and uniform random weight from $\{1,\ldots,W\}$, where $W = O(d)$. We show that there exists an LCA that is consistent with an exact MST of $G$, with probe complexity $\tilde{O}(\sqrt{n}d^2)$.

翻译：本文研究用于构建生成树的局部计算算法。在此设定下，目标是对输入图 \( G \) 中的每条边 \( e \in E \) 进行局部判定，确定其是否属于由 \( G \) 及算法随机性隐式定义的生成树 \( T \)。已知在一般图（即使是简单图族）中，生成树的局部计算算法并不存在。我们识别出一类经过充分研究的自然图类——膨胀图——确实允许生成树的亚线性时间局部计算算法。这一结论或许令人惊讶，因为先前关于膨胀图的研究仅成功设计了稀疏生成子图的局部计算算法，而非完整的生成树。对于传导率至少为 \( φ \)、最大度至多为 \( d \)（不必为常数）的图，我们设计了一个探测复杂度为 \( O\left(\sqrt{n}\left(\frac{\log^2 n}{φ^2} + d\right)\right) \) 的局部计算算法，当 \( φ \) 和 \( d \) 为常数时该复杂度近乎最优，因为即使在膨胀图上也需要 \( Ω(\sqrt{n}) \) 次探测。接下来，我们证明对于自然的随机图类 \( G(n, p) \)（其中 \( np = n^δ \)，\( δ > 0 \) 为任意常数，这类图以高概率为膨胀图），可以突破 \( \sqrt{n} \) 的下界限制。具体而言，我们为此类图设计了一个平均情况下的局部计算算法，其探测复杂度为 \( \tilde{O}(\sqrt{n^{1 - δ}}) \)。最后，我们将技术扩展至加权膨胀图上的最小生成树问题。具体地，给定一个具有足够强膨胀性的 \( d \)-正则无权图 \( \bar{G} \)，考虑通过为每条边独立且均匀地从 \( \{1,\ldots,W\} \)（其中 \( W = O(d) \)）中分配随机权重得到的加权图 \( G \)。我们证明存在一个与 \( G \) 的精确最小生成树保持一致的局部计算算法，其探测复杂度为 \( \tilde{O}(\sqrt{n}d^2) \)。