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 @math -coloring algorithm running on the Congested Clique which has an expected running time of (i) @math rounds, if @math ; and (ii) @math rounds otherwise. Applying the simulation theorem to the Congested-Clique @math -coloring algorithm yields an @math -round @math -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 @math model), the major impediment to constructing fast algorithms is the @math 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.
The earliest interesting algorithm in the Congested Clique model is an @math -round deterministic algorithm to compute a minimum spanning tree (MST), due to @cite_12 . @cite_1 presented a random @math -round algorithm in the Congested Clique model that produced a constant-factor approximation algorithm for the metric facility location problem. @cite_6 @cite_9 considered the more-general non-uniform metric facility location in the Congested Clique model and presented a constant-factor approximation running in expected @math rounds. reduce the metric facility location problem to the problem of computing a @math -ruling set of a spanning subgraph of the underlying communication network and show how to solve this in @math rounds in expectation. In 2013, Lenzen presented a routing protocol to solve a problem called an @cite_7 . The setup for this problem is that each node @math is given a set of @math messages, each of size @math , @math , with destinations @math , @math . Messages are globally lexicographically ordered by their source @math , destination @math , and @math . Each node is also the destination of at most @math messages. Lenzen's routing protocol solves the Information Distribution Task in @math rounds.
@cite_7 @cite_9 @cite_1 @cite_6 @cite_12
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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 (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 almost-tight bounds on the message complexity of these problems. Specifically, we show that Ω(n2) 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 algorithm 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 constant-time and near-constant-time distributed algorithms for a variety of problems in the congested clique model. We show how to compute a 3-ruling set in expected O(logloglogn) rounds and using this, we obtain a constant-approximation to metric facility location, also in expected O(logloglogn) rounds. 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. We study the broadcast version of the CONGEST CLIQUE model of distributed computing. In this model, in each round, any node in a network of size @math can send the same message (i.e. broadcast a message) of limited size to every other node in the network. Nanongkai presented in [STOC'14] a randomized @math -approximation algorithm to compute all pairs shortest paths (APSP) in time @math on weighted graphs, where we use the convention that @math is essentially @math polylog @math and @math is essentially @math polylog @math . We complement this result by proving that any randomized @math -approximation of APSP and @math -approximation of the diameter of a graph takes @math time in the worst case. This demonstrates that getting a negligible improvement in the approximation factor requires significantly more time. Furthermore this bound implies that already computing a @math -approximation of all pairs shortest paths is among the hardest graph-problems in the broadcast-version of the CONGEST CLIQUE model and contrasts a recent @math -approximation for APSP that runs in time @math in the unicast version of the CONGEST CLIQUE model. On the positive side we provide a deterministic version of Nanongkai's @math -approximation algorithm for APSP. To do so we present a fast deterministic construction of small hitting sets. We also show how to replace another randomized part within Nanongkai's algorithm with a deterministic source-detection algorithm designed for the CONGEST model presented by Lenzen and Peleg at PODC'13. In this work, we use algebraic methods for studying distance computation and subgraph detection tasks in the congested clique model. Specifically, we adapt parallel matrix multiplication implementations to the congested clique, obtaining an @math round matrix multiplication algorithm, where @math is the exponent of matrix multiplication. In conjunction with known techniques from centralised algorithmics, this gives significant improvements over previous best upper bounds in the congested clique model. The highlight results include: -- triangle and 4-cycle counting in @math rounds, improving upon the @math triangle detection algorithm of [DISC 2012], -- a @math -approximation of all-pairs shortest paths in @math rounds, improving upon the @math -round @math -approximation algorithm of Nanongkai [STOC 2014], and -- computing the girth in @math rounds, which is the first non-trivial solution in this model. In addition, we present a novel constant-round combinatorial algorithm for detecting 4-cycles.