Fast Spanning Tree Sampling in Broadcast Congested Clique

TL;DR

This paper introduces the first polylogarithmic-round algorithm for efficiently sampling near-uniform random spanning trees in the broadcast congested clique model, significantly improving over previous methods.

Contribution

It provides the first polylogarithmic-round distributed algorithm for approximate uniform spanning tree sampling in the congested clique model, with strong theoretical guarantees.

Findings

Achieves sampling within $O(n^{-c})$ total variation distance.

Runs in $c imes ext{polylog}(n)$ rounds, for any constant $c>0$.

Exponential improvement over previous algorithms.

Abstract

We present the first polylogarithmic-round algorithm for sampling a random spanning tree in the (Broadcast) Congested Clique model. For any constant $c > 0$, our algorithm outputs a sample from a distribution whose total variation distance from the uniform spanning tree distribution is at most $O(n^{-c})$ in at most $c \cdot \log^{O(1)}(n)$ rounds. The exponent hidden in $\log^{O(1)}(n)$ is an absolute constant independent of $c$ and $n$. This is an exponential improvement over the previous best algorithm of Pemmaraju, Roy, and Sobel (PODC 2025) for the Congested Clique model.