# Distributed Computation in the Node-Congested Clique

The Congested Clique model of distributed computing, which was introduced by Lotker, Patt-Shamir, Pavlov, and Peleg [SPAA'03, SICOMP'05] and was motivated as "a 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( 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( n) many messages of size O( 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.

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