Distributed Computation in the Node-Congested Clique.

The Congested Clique model of distributed computing, which was introduced by Lotker, Patt-Shamir, Pavlov, and Peleg [SPAAu002703, SICOMPu002705] and was motivated as simple model for overlay networks, has received extensive attention over the past few years. In this model, nodes of the system are connected as a clique and can communicate in synchronous rounds, where per round each node can send $O(log n)$ bits to each other node, all the same time. The fact that this model allows each node to send and receive a linear number of messages at the same time seems to limit the relevance of the model for overlay networks. Towards addressing this issue, in this paper, we introduce the Node-Congested Clique as a general communication network model. Similarly to the Congested Clique model, the nodes are connected as a clique and messages are sent in synchronous communication rounds. However, here, per round, every node can send and receive only $O(log n)$ many messages of size $O(log n)$. To initiate research on our network model, we present distributed algorithms for the Minimum Spanning Tree, BFS Tree, Maximal Independent Set, Maximal Matching, and Coloring problem for an input graph $G=(V,E)$, where each clique node initially only knows a single node of $G$ and its incident edges. For the Minimum Spanning Tree problem, our runtime is polylogarithmic. In all other cases the runtime of our algorithms mainly depends on the arboricity $a$ of $G$, which is a constant for many important graph families such as planar graphs. At the core of these algorithms is a distributed algorithm that assigns directions to the edges of $G$ so that at the end, every node is incident to at most $O(a)$ outgoing edges.