  The congested clique is a synchronous, message-passing model of distributed
computing in which each computational unit (node) in each round can send
message of O(log n) bits to each other node of the network, where n is the
number 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 the
broadcast 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 maximum
number of different messages a node can send in one round. Following recent
progress in design of algorihms for graph connectivity and minimum span- ning
forest (MSF) in the unicast congested clique, we study these problems in
limited variants of the congested clique. We present the first sub-logarithmic
algorithm for connected components in the broadcast congested clique. Then, we
show that efficient unicast deterministic algorithm for MSF and randomized
algorithm for connected components can be efficiently imple- mented in the
rcast model with range r = 2, the weakest model of the congested clique above
the broadcast variant (r = 1) in the hierarchy with respect to range. More
importantly, our al- gorithms give the first solutions with optimal capacity of
communication edges, while preserving small round complexity.
