# Random Sampling Applied to the MST Problem in the Node Congested Clique Model

The Congested Clique model, proposed by Lotker et al. [SPAA'03, SICOMP'05], was introduced in order to provide a simple abstraction for overlay networks. Congested Clique is a model of distributed (or parallel) computing, in which there are n players (nodes) with unique identifiers from set 1, ..., n, which perform computations in synchronous rounds. Each round consists of the phase of unlimited local computation and the communication phase. While communicating, each pair of nodes is allowed to exchange a single message of size O( n) bits. Since, in a single round, each player can communicate with even Θ(n) other players, the model seems to be to powerful to imitate bandwidth restriction emerging from the underlying network. In this paper we study a restricted version of the Congested Clique model, the Node Congested Clique model, proposed by Augustine et al. [arxiv1805]. The additional restriction is that in a single communication phase, a player is allowed to send / receive only O( n) messages. In this paper, we provide communication primitives that improve the round complexity of the MST (Minimum Spanning Tree) algorithm by Augustine et al. [arxiv1805] to O(^3 n) rounds. Moreover, we propose a different approach to this problem that requires only O(^3 n / n) rounds, and has smaller dependence on the weights of the edges. Besides the faster MST algorithm, we consider the key contributions to be: - an efficient implementation of some basic protocols, - a tighter analysis of a special case of the sampling approach by Karger, Klein and Tarjan [JACM'95] (and related results by Pemmaraju and Sardeshmukh [FSTTCS'16]), - an application of sparse recovery techniques going slightly beyond the standard usage of linear graph sketching by Ahn, Guha and McGregor [SODA'12]

READ FULL TEXT