Minimum-weight Spanning Tree Construction in $O(log log log n)$ Rounds on the Congested Clique

This paper considers the \textit{minimum spanning tree (MST)} problem in the Congested Clique model and presents an algorithm that runs in $O(\log \log \log n)$ rounds, with high probability. Prior to this, the fastest MST algorithm in this model was a deterministic algorithm due to Lotker et al.(SIAM J on Comp, 2005) from about a decade ago. A key step along the way to designing this MST algorithm is a \textit{connectivity verification} algorithm that not only runs in $O(\log \log \log n)$ rounds with high probability, but also has low message complexity. This allows the fast computation of an MST by running multiple instances of the connectivity verification algorithm in parallel. These results depend on a new edge-sampling theorem, developed in the paper, that says that if each edge $e = {u, v}$ is sampled independently with probability $c \log^2 n/\min{\mbox{degree}(u), \mbox{degree}(v)}$ (for a large enough constant $c$) then all cuts of size at least $n$ are approximated in the sampled graph. This sampling theorem is inspired by series of papers on graph sparsification via random edge sampling due to Karger(STOC 1994), Bencz'ur and Karger~(STOC 1996, arxiv 2002), and Fung et al.~(STOC 2011). The edge sampling techniques in these papers use probabilities that are functions of edge-connectivity or a related measure called edge-strength. For the purposes of this paper, these edge-connectivity measures seem too costly to compute and the main technical contribution of this paper is to show that degree-based edge-sampling suffices to approximate large cuts.

💡 Research Summary

The paper tackles the Minimum Spanning Tree (MST) problem in the Congested Clique model, where every node can communicate directly with every other node but each communication link can carry only O(log n) bits per round. Prior to this work, the fastest known algorithm for MST in this setting was a deterministic O(log log n)‑round algorithm by Lotker et al. (2005). The authors present a randomized algorithm that computes an MST in O(log log log n) rounds with high probability, thereby achieving an exponential speed‑up over the previous state of the art.

The central technical contribution is a new edge‑sampling theorem that uses only local degree information. For each edge e = {u, v}, the algorithm samples e independently with probability

 pₑ = min{c·log² n / min{deg(u), deg(v)}, 1},

where c is a sufficiently large constant and deg(·) denotes the vertex degree in the original graph. This “degree‑based” sampling yields two crucial properties with high probability: (i) the sampled subgraph ˆG contains only O(n·log² n) edges, and (ii) every cut of size at least n in the original graph retains at least one edge in ˆG. Consequently, large cuts are essentially preserved while the graph becomes sparse enough to be handled efficiently in the Congested Clique.

Using the sparsified graph, the authors design a Connectivity Verification subroutine that runs in O(log log log n) rounds. Each node initially knows its incident edges in ˆG, computes a component identifier (the minimum node ID in its component), and then exchanges information about inter‑component edges. Lenzen’s deterministic routing protocol allows each node to send O(n) messages in constant time, ensuring that the communication bottleneck does not dominate the round complexity. After a constant number of such exchanges, every node learns a maximal spanning forest of ˆG, which suffices to decide whether the original graph is connected.

The MST algorithm builds on this connectivity subroutine by executing multiple instances in parallel, mimicking Borůvka’s method. In each iteration, the lightest inter‑component edge (according to the original weights) is selected, components are merged, and the process repeats. Because each iteration can be performed within O(log log log n) rounds, and the number of iterations needed is only O(log log log n) (due to the exponential reduction in the number of components), the total round complexity remains O(log log log n) with high probability. The algorithm also maintains low message complexity: the sampling phase sends O(n·log² n) messages, and each connectivity verification phase exchanges only O(1) messages per component.

Beyond MST, the paper’s degree‑based sampling technique offers a practical alternative to the edge‑strength or edge‑connectivity based sampling used in classic sparsification results by Karger, Benczúr‑Karger, and Fung et al. Those earlier methods require global graph properties that are expensive to compute in distributed settings. By relying solely on locally available degree information, the new approach is well‑suited to the Congested Clique’s bandwidth constraints and can be adapted to other distributed graph problems such as cut approximation, flow computation, and general graph sparsification.

In summary, the authors achieve an O(log log log n)‑round MST algorithm for the Congested Clique, introduce a novel degree‑based edge‑sampling theorem that preserves large cuts while drastically reducing edge count, and demonstrate how to combine these tools with Lenzen’s routing protocol to overcome congestion bottlenecks. This work sets a new benchmark for round complexity in the Congested Clique and opens avenues for applying degree‑based sparsification to a broader class of distributed graph algorithms.