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

The Congested Clique model, proposed by Lotker et al. [SPAAu002703, SICOMPu002705], 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 $mathcal{O}(log n)$ bits. Since, in a single round, each player can communicate with even $Theta(n)$ other players, the model seems to be to powerful to imitate bandwidth restriction emerging from the underlying network. 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 $mathcal{O}(log n)$ 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 $mathcal{O}(log^3 n)$ rounds. Moreover, we propose a different approach to this problem that requires only $mathcal{O}(log^3 n / log log n)$ 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 [JACMu002795] (and related results by Pemmaraju and Sardeshmukh [FSTTCSu002716]), - an application of sparse recovery techniques going slightly beyond the standard usage of linear graph sketching by Ahn, Guha and McGregor [SODAu002712]