We study two fundamental graph problems, Graph Connectivity (GC) and Minimum Spanning Tree (MST), in the well-studied Congested Clique model, and present several new bounds on the time and message complexities of randomized algorithms for these problems. No non-trivial (i.e., super-constant) time lower bounds are known for either of the aforementioned problems; in particular, an important open question is whether or not constant-round algorithms exist for these problems. We make progress toward answering this question by presenting randomized Monte Carlo algorithms for both problems that run in O(log log log n) rounds (where n is the size of the clique). Our results improve by an exponential factor on the long-standing (deterministic) time bound of O(log log n) rounds for these problems due to Lotker et al. (SICOMP 2005). Our algorithms make use of several algorithmic tools including graph sketching, random sampling, and fast sorting.The second contribution of this paper is to present several almosttight bounds on the message complexity of these problems. Specifically, we show that Ω(n 2 ) messages are needed by any algorithm (including randomized Monte Carlo algorithms, and regardless of the number of rounds) that solves the GC (and hence also the MST) problem if each machine in the Congested Clique has initial knowledge only of itself (the so-called KT0 model). In contrast, if the machines have initial knowledge of their neighbors' IDs (the so-called KT1 model), we present a randomized Monte Carlo algorithm for MST that uses O(n polylog n) messages and runs in O(polylog n) rounds. To complement this, we also present a lower bound in the KT1 model that shows that Ω(n) messages are required by any al- * gorithm that solves GC, regardless of the number of rounds used. Our results are a step toward understanding the power of randomization in the Congested Clique with respect to both time and message complexity.

This paper presents a distributed O(1)-approximation algorithm, with expected-O(log log n) running time, in the CON GE ST model for the metric facility location problem on a size-n clique network. Though metric facility location has been considered by a number of researchers in low-diameter settings, this is the first sub-logarithmic-round algorithm for the problem that yields an O(1)-approximation in the setting of non-uniform facility opening costs. In order to obtain this result, our paper makes three main technical contributions. First, we show a new lower bound for metric facility location, extending the lower bound of Bȃdoiu et al. (ICALP 2005) that applies only to the special case of uniform facility opening costs. Next, we demonstrate a reduction of the distributed metric facility location problem to the problem of computing an O(1)-ruling set of an appropriate spanning subgraph. Finally, we present a sub-logarithmic-round (in expectation) algorithm for computing a 2-ruling set in a spanning subgraph of a clique. Our algorithm accomplishes this by using a combination of randomized and deterministic sparsification.

This paper presents constant-time and near-constant-time distributed algorithms for a variety of problems in the congested clique model. We show how to compute a 2-ruling set in O(log log log n) rounds with high probability and using this, we obtain a constant-approximation to metric facility location, also in O(log log log n) rounds with high probability. In addition, assuming an input metric space of constant doubling dimension, we obtain constant-round algorithms to compute constant-factor approximations to the minimum spanning tree and the metric facility location problems. These results significantly improve on the running time of the fastest known algorithms for these problems in the congested clique setting.

The main results of this paper are (I) a simulation algorithm which, under quite general constraints, transforms algorithms running on the Congested Clique into algorithms running in the MapReduce model, and (II) a distributed O(∆)-coloring algorithm running on the Congested Clique which has an expected running time of O(1) rounds, if ∆ ≥ Θ(log 4 n); and O(log log log n) rounds otherwise. Applying the simulation theorem to the Congested Clique O(∆)-coloring algorithm yields an O(1)-round O(∆)-coloring algorithm in the MapReduce model.Our simulation algorithm illustrates a natural correspondence between per-node bandwidth in the Congested Clique model and memory per machine in the MapReduce model. In the Congested Clique (and more generally, any network in the CON GE ST model), the major impediment to constructing fast algorithms is the O(log n) restriction on message sizes. Similarly, in the MapReduce model, the combined restrictions on memory per machine and total system memory have a dominant effect on algorithm design. In showing a fairly general simulation algorithm, we highlight the similarities and differences between these models.

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