Connectivity and Minimum Cut Approximation in the Broadcast Congested Clique

Abstract

In this paper we present two graph algorithms in the Broadcast Congested Clique model. In this model, there are n players, which communicate in synchronous rounds. Each player represents a single node of the input graph; initially each player knows the set of edges incident to his node. In each round of communication each node can broadcast a single b–bit message to all other nodes; usually $b \in {\mathcal {O}}(\log n)$. The goal is to compute some function of the input graph.

The first result we present is the first sub-logarithmic deterministic algorithm finding a maximal spanning forest of an n node graph in the Broadcast Congested Clique, which requires only ${\mathcal {O}}(\log n / \log \log n)$ rounds. The second result is a randomized $1 + \epsilon$ approximation algorithm finding the minimum cut of an n node graph, which requires only ${\mathcal {O}}(\log n)$ maximal spanning forest computations. In the Broadcast Congested Clique this approach, combined with the new maximal spanning forest algorithm, yields an ${\mathcal {O}}(\log ^2 n / \log \log n)$ round algorithm. Additionally, it may be applied to different models, i.e. in the multi-pass semi-streaming model it allows to reduce required memory by $\varTheta (\log n)$ factor, with only ${\mathcal {O}}(\log ^* n)$ passes over the data stream.

This work was supported by the National Science Centre, Poland grant 2017/25/B/ST6/02010. Results from Sect. 3 were presented as Brief Announcement at DISC 2018 [8] and their initial version was obtained with support from the National Science Centre, Poland grant DEC-2012/07/B/ST6/01534.