# A Deterministic Algorithm for the MST Problem in Constant Rounds of Congested Clique

In this paper, we show that the Minimum Spanning Tree problem can be solved deterministically, in O(1) rounds of the CongestedClique model. In the CongestedClique model, there are n players that perform computation in synchronous rounds. Each round consist of a phase of local computation and a phase of communication, in which each pair of players is allowed to exchange O(log n) bit messages. The studies of this model began with the MST problem: in the paper by Lotker et al.[SPAA'03, SICOMP'05] that defines the CongestedClique model the authors give a deterministic O(loglog n) round algorithm that improved over a trivial O(log n) round adaptation of Borůvka's algorithm. There was a sequence of gradual improvements to this result: an O(logloglog n) round algorithm by Hegeman et al. [PODC'15], an O(log^* n) round algorithm by Ghaffari and Parter, [PODC'16] and an O(1) round algorithm by Jurdziński and Nowicki, [SODA'18], but all those algorithms were randomized, which left the question about the existence of any deterministic o(loglog n) round algorithms for the Minimum Spanning Tree problem open. Our result resolves this question and establishes that O(1) rounds is enough to solve the MST problem in the CongestedClique model, even if we are not allowed to use any randomness. Furthermore, the amount of communication needed by the algorithm makes it applicable to some variants of the MPC model.

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