We study the minimum spanning tree (MST) problem in the massively parallel computation (MPC) model. Our focus is particularly on the *strictly sublinear* regime of MPC where the space per machine is $O(n^\delta)$. Here $n$ is the number of vertices and constant $\delta \in (0, 1)$ can be made arbitrarily small. The MST problem admits a simple and folklore $O(\log n)$-round algorithm in the MPC model. When the weights can be arbitrary, this matches a conditional lower bound of $\Omega(\log n)$ which follows from a well-known 1vs2-Cycle conjecture. As such, much of the literature focuses on breaking the logarithmic barrier in more structured variants of the problem, such as when the vertices correspond to points in low- [ANOY14, STOC'14] or high-dimensional Euclidean spaces [JMNZ, SODA'24]. In this work, we focus more generally on metric spaces. Namely, all pairwise weights are provided and guaranteed to satisfy the triangle inequality, but are otherwise unconstrained. We show that for any $\varepsilon > 0$, a $(1+\varepsilon)$-approximate MST can be found in $O(\log \frac{1}{\varepsilon} + \log \log n)$ rounds, which is the first $o(\log n)$-round algorithm for finding any constant approximation in this setting. Other than being applicable to more general weight functions, our algorithm also slightly improves the $O(\log \log n \cdot \log \log \log n)$ round-complexity of [JMNZ24, SODA'24] and significantly improves its approximation from a large constant to $1+\varepsilon$. On the lower bound side, we prove that under the 1vs2-Cycle conjecture, $\Omega(\log \frac{1}{\varepsilon})$ rounds are needed for finding a $(1+\varepsilon)$-approximate MST in general metrics. It is worth noting that while many existing lower bounds in the MPC model under the 1vs2-Cycle conjecture only hold against "component stable" algorithms, our lower bound applies to *all* algorithms.

翻译：我们研究大规模并行计算（MPC）模型中的最小生成树（MST）问题，特别关注MPC的*严格亚线性*区域，其中每台机器的空间为$O(n^\delta)$。此处$n$为顶点数，常数$\delta \in (0, 1)$可任意小。MST问题在MPC模型中存在一个简单且广为人知的$O(\log n)$轮算法。当权重可任意取值时，该复杂度与基于著名1vs2环猜想推导出的条件性下界$\Omega(\log n)$相匹配。因此，现有研究多致力于在更具结构性的问题变体中突破对数障碍，例如当顶点对应于低维[ANOY14, STOC'14]或高维欧几里得空间[JMNZ, SODA'24]中的点时。本文更一般地关注度量空间：即所有成对权重均已知且保证满足三角不等式，但除此之外无其他约束。我们证明对于任意$\varepsilon > 0$，可在$O(\log \frac{1}{\varepsilon} + \log \log n)$轮内找到$(1+\varepsilon)$近似MST，这是在该设定下首个实现常数近似的$o(\log n)$轮算法。除了适用于更一般的权重函数外，我们的算法还略微改进了[JMNZ24, SODA'24]中$O(\log \log n \cdot \log \log \log n)$的轮复杂度，并将其近似比从大常数显著提升至$1+\varepsilon$。在下界方面，我们证明基于1vs2环猜想，在一般度量空间中寻找$(1+\varepsilon)$近似MST需要$\Omega(\log \frac{1}{\varepsilon})$轮。值得注意的是，虽然MPC模型中许多基于1vs2环猜想的现有下界仅针对"组件稳定"算法成立，但我们的下界适用于*所有*算法。