2016/10/12 by Sriram V. Pemmaraju, Pemmaraju, Sriram V., Vivek B. Sardeshmukh +1
Computer Science · #Algorithms and Data Compression #Complexity and Algorithms in Graphs #Distributed #Distributed systems and fault tolerance #FOS: Computer and information sciences #Parallel #and Cluster Computing (cs.DC)
paper · pdf · doi:10.48550/arxiv.1610.03897
openalex publication_date 2016/10/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 log n) from O(log log n) (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 O(√(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 O(n1 + ε/ε) messages. Setting ε = loglog n/log n leads to the first sub-logarithmic round Congested Clique MST algorithm that uses only O(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(log n)-wise-independent linear graph sketches (Cormode and Firmani, Dist.~Par.~Databases, 2014) for generating MST candidate edges.