# What Can We Compute in a Single Round of the Congested Clique?

We study the computational power of one-round distributed algorithms in the congested clique model. We show that any one-round algorithm that computes a minimum spanning tree (MST) in the unicast congested clique must use a link bandwidth of Ω(log^3 n) bits in the worst case. This is the first round complexity lower bound in the unicast congested clique for a problem where the output size is small, i.e., O(nlog n) bits. Our main technical contribution is to investigate one-round algorithms in the broadcast congested clique and, equivalently, the distributed graph sketching model where the nodes send their message to a referee who computes the output. First, we present a tight lower bound of Ω(n) bits for the message size of computing a breadth-first search tree. Then, we prove that computing a k-edge connected spanning subgraph (k-ECSS) requires messages of size at least Ω( klog^2(n/k) ). We also show that verifying whether a given vertex coloring forms a weak 2-coloring of the input graph requires messages of Ω(n^1/3log^2/3n) bits, and the same lower bound holds for verifying whether a subset of nodes forms a maximal independent set or a minimal dominating set. Interestingly, it turns out that the same class of lower bound graphs for the distributed sketching model is versatile enough to yield a space lower bound of Ω(n^2) bits for verifying symmetry breaking problems such as weak 2-coloring in the fully dynamic turnstile model, where the input arrives as a stream of edges. We also (nearly) settle the space complexity of the k-ECSS problem in the streaming model by extending the work of Kapralov et al. (FOCS 2017): We prove a communication complexity lower bound for a direct sum variant of the UR_k^⊂ problem and show that this implies Ω(k nlog^2(n/k)) bits of memory for computing a k-ECSS.

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