What Can We Compute in a Single Round of the Congested Clique?

TL;DR

This paper establishes a fundamental lower bound on the bandwidth and rounds needed for computing a minimum spanning tree in the unicast congested clique model, highlighting inherent communication complexity constraints.

Contribution

It proves the first round complexity lower bound for MST in the unicast congested clique with small output size, under standard message size assumptions.

Findings

One-round MST computation requires ( extsuperscript{3}) bits bandwidth in the worst case.

At least two rounds are necessary for MST with O( extsuperscript{log} n) message size.

Lower bounds apply even when each MST edge is output by an incident node.

Abstract

We show that any one-round algorithm that computes a minimum spanning tree (MST) in the unicast congested clique must use a link bandwidth of $\Omega(\log^3 n)$ bits in the worst case. Consequently, computing an MST under the standard assumption of $O(\log n)$-size messages requires at least $2$ rounds. This is the first round complexity lower bound in the unicast congested clique for a problem where the output size is small, i.e., $O(n\log n)$ bits. Our lower bound holds as long as every edge of the MST is output by an incident node. To the best of our knowledge, all prior lower bounds for the unicast congested clique either considered problems with large output sizes (e.g., triangle enumeration) or required every node to learn the entire output.