Massively Parallel Minimum Spanning Tree in General Metric Spaces
Amir Azarmehr, Soheil Behnezhad, Rajesh Jayaram, Jakub Łącki, Vahab Mirrokni, Peilin Zhong
TL;DR
This work proves that in the strictly sublinear MPC regime, MST in general metric spaces can be computed to a (1+ε) approximation in O(log(1/ε) + log log n) rounds, using per-machine space n^δ and near-linear total space. The algorithm combines an MPX-based low-diameter partition hierarchy with a Borůvka-style merge that avoids constructing all intermediate partitions, achieving subloground complexity previously known only for specialized metric settings. A matching conditional lower bound under the 1vs2-Cycle conjecture shows that the ε-dependence is optimal and that no o(log(1/ε))-round sublogarithmic MPC algorithm exists for general metrics, even for (1,2)-metrics, unless the conjecture is false. The results extend to TSP via known reductions, and they advance understanding of MPC lower bounds beyond component-stable models by introducing a random relabeling technique. Overall, the paper delivers a near-optimal sublogarithmic MPC MST algorithm in general metric spaces and a robust conditional lower bound, with broad implications for distributed graph optimization in MPC.
Abstract
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 $δ\in (0, 1)$ 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 $Ω(\log n)$ 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 [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, $Ω(\log \frac{1}{\varepsilon})$ 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.