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The congested clique is a synchronous, message-passing model of distributedcomputing in which each computational unit (node) in each round can sendmessage of O(log n) bits to each other node of the network, where n is thenumber of nodes. This model has been considered under two extreme scanarios:unicast or broadcast. In the unicast model, a node can send (possibly)different message to each other node of the network. In contrast, in thebroadcast model each node sends a single (the same) message to all other nodes.We study the congested clique model parametrized by the range r, the maximumnumber of different messages a node can send in one round. Following recentprogress in design of algorihms for graph connectivity and minimum span- ningforest (MSF) in the unicast congested clique, we study these problems inlimited variants of the congested clique. We present the first sub-logarithmicalgorithm for connected components in the broadcast congested clique. Then, weshow that efficient unicast deterministic algorithm for MSF and randomizedalgorithm for connected components can be efficiently imple- mented in thercast model with range r = 2, the weakest model of the congested clique abovethe broadcast variant (r = 1) in the hierarchy with respect to range. Moreimportantly, our al- gorithms give the first solutions with optimal capacity ofcommunication edges, while preserving small round complexity.

MSF and Connectivity in Limited Variants of the Congested Clique