MSF and Connectivity in Limited Variants of the Congested Clique

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. Following [1], 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 spanning 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 [11] and randomized algorithm for connected components [5] can be efficiently implemented 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 algorithms give the first solutions with optimal capacity of communication edges, while preserving small round complexity.