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We present a distributed randomized algorithm finding Minimum Spanning Tree (MST) of a given graph in O(1) rounds, with high probability, in the congested clique model.The input graph in the congested clique model is a graph of n nodes, where each node initially knows only its incident edges. The communication graph is a clique with limited edge bandwidth: each two nodes (not necessarily neighbours in the input graph) can exchange O(log n) bits.As in previous works, the key part of the MST algorithm is an efficient Connected Components (CC) algorithm. However, unlike the former approaches, we do not aim at simulating the standard Boruvka's algorithm, at least at initial stages of the CC algorithm. Instead, we develop a new technique which combines connected components of sample sparse subgraphs of the input graph in order to accelerate the process of uncovering connected components of the original input graph. More specifically, we develop a sparsification technique which reduces an initial CC problem in O(1) rounds to its two restricted instances. The former instance has a graph with maximal degree

We provide a simple new randomized contraction approach to the global minimum cut problem for simple undirected graphs. The contractions exploit 2-out edge sampling from each vertex rather than the standard uniform edge sampling. We demonstrate the power of our new approach by obtaining better algorithms for sequential, distributed, and parallel models of computation. Our end results include the following randomized algorithms for computing edge connectivity, with high probability 1 :• Two sequential algorithms with complexities O(m log n) and O(m+n log 3 n). These improve on a long line of developments including a celebrated O(m log 3 n) algorithm of Karger [STOC'96] and the state of the art O(m log 2 n(log log n) 2 ) algorithm of Henzinger et al. [SODA'17]. Moreover, our O(m + n log 3 n) algorithm is optimal when m = Ω(n log 3 n).• AnÕ(n 0.8 D 0.2 + n 0.9 ) round distributed algorithm, where D denotes the graph diameter. This improves substantially on a recent breakthrough of Daga et al. [STOC'19], which achieved a round complexity ofÕ(n 1−1/353 D 1/353 + n 1−1/706 ), hence providing the first sublinear distributed algorithm for exactly computing the edge connectivity.• The first O(1) round algorithm for the massively parallel computation setting with linear memory per machine.

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