Super-Fast MST Algorithms in the Congested Clique Using o(m) Messages

In a sequence of recent results (PODC 2015 and PODC 2016), the running time of the fastest algorithm for the minimum spanning tree (MST) problem in the Congested Clique model was first improved to O(log log logn) from O(log logn) (Hegeman et al., PODC 2015) and then to O(log∗ n) (Ghaffari and Parter, PODC 2016). All of these algorithms use Θ(n2) messages independent of the number of edges in the input graph. This paper positively answers a question raised in Hegeman et al., and presents the first “super-fast” MST algorithm with o(m) message complexity for input graphs with m edges. Specifically, we present an algorithm running in O(log∗ n) rounds, with message complexity Õ( √ m · n) and then build on this algorithm to derive a family of algorithms, containing for any ε, 0 < ε ≤ 1, an algorithm running in O(log∗ n/ε) rounds, using Õ(n1+ε/ε) messages. Setting ε = log logn/ logn leads to the first sub-logarithmic round Congested Clique MST algorithm that uses only Õ(n) messages. Our primary tools in achieving these results are (i) a component-wise bound on the number of candidates for MST edges, extending the sampling lemma of Karger, Klein, and Tarjan (Karger, Klein, and Tarjan, JACM 1995) and (ii) Θ(logn)-wiseindependent linear graph sketches (Cormode and Firmani, Dist. Par. Databases, 2014) for generating MST candidate edges.