Massively Parallel Minimum Spanning Tree in General Metric Spaces

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 δ ). Here n is the number of vertices and constant δ ∈ (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 Ω(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 [JMNZ24, 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 ε > 0, a (1 + ε)-approximate MST can be found in O(log 1 ε + 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 • log log log n) round-complexity of [JMNZ24, SODA'24] and significantly improves its approximation from a large constant to 1 + ε.

On the lower bound side, we prove that under the 1vs2-Cycle conjecture, Ω(log 1 ε ) rounds are needed for finding a (1 + ε)-approximate MST in general metrics. This implies that (i) the ε-dependency of our algorithm is optimal, (ii) it is necessary to approximate MST in order to beat Ω(log n) rounds in the metric case, and (iii) computing metric MST is strictly harder than computing MST in low-dimensional Euclidean spaces.

It is also 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. Indeed, a conceptual contribution of our paper is to provide a natural way of lifting the component stability assumption which we hope to have other applications.