MIS in the Congested Clique Model in $O(\log \log \Delta)$ Rounds

TL;DR

This paper presents a new distributed algorithm for computing a maximal independent set in the congested clique model that operates in $O( ext{log log} \Delta)$ rounds, significantly improving previous bounds.

Contribution

The authors develop a two-stage MIS algorithm that first reduces the graph's degree rapidly and then applies existing methods, achieving a faster overall runtime.

Findings

Achieves MIS in $O( ext{log log} \Delta)$ rounds in the congested clique.

Introduces a novel simulation of sequential greedy iterations in the distributed setting.

Reduces the maximum degree of the residual graph to poly-logarithmic in the first stage.

Abstract

We give a maximal independent set (MIS) algorithm that runs in $O(\log \log \Delta)$ rounds in the congested clique model, where $\Delta$ is the maximum degree of the input graph. This improves upon the $O(\frac{\log(\Delta) \cdot \log \log \Delta}{\sqrt{\log n}} + \log \log \Delta )$ rounds algorithm of [Ghaffari, PODC '17], where $n$ is the number of vertices of the input graph. In the first stage of our algorithm, we simulate the first $O(\frac{n}{\text{poly} \log n})$ iterations of the sequential random order Greedy algorithm for MIS in the congested clique model in $O(\log \log \Delta)$ rounds. This thins out the input graph relatively quickly: After this stage, the maximum degree of the residual graph is poly-logarithmic. In the second stage, we run the MIS algorithm of [Ghaffari, PODC '17] on the residual graph, which completes in $O(\log \log \Delta)$ rounds on graphs of poly-logarithmic degree.