# MIS in the Congested Clique Model in O( Δ) Rounds

We give a maximal independent set (MIS) algorithm that runs in O(Δ) rounds in the congested clique model, where Δ is the maximum degree of the input graph. This improves upon the O((Δ) ·Δ/√( n) + Δ ) 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(n/poly n) iterations of the sequential random order Greedy algorithm for MIS in the congested clique model in O(Δ) 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(Δ) rounds on graphs of poly-logarithmic degree.

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