We present simple deterministic algorithms for subgraph finding and enumeration in the broadcast CONGEST model of distributed computation: -- For any constant @math , detecting @math -paths and trees on @math nodes can be done in constantly many rounds, and @math -cycles in @math rounds. -- On @math -degenerate graphs, cliques and @math -cycles can be enumerated in @math rounds, and @math -cycles in @math rounds. In many cases, these bounds are tight up to logarithmic factors. Moreover, we show that the algorithms for @math -degenerate graphs can be improved to optimal complexity @math and @math , respectively, in the supported CONGEST model, which can be seen as an intermediate model between CONGEST and the congested clique.
To our knowledge, no other subgraph detection lower bounds are known in the model. In particular, proving lower bounds for triangle detection seems to be a particularly difficult challenge. However, for the broadcast congested clique model, @cite_24 give an @math round lower bound, which also applies to the broadcast model. Moreover, for triangle enumeration lower bounds are known, also in the stronger congested clique model @cite_12 @cite_25 .
@cite_24 @cite_25 @cite_12
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We study the computation power of the congested clique, a model of distributed computation where n players communicate with each other over a complete network in order to compute some function of their inputs. The number of bits that can be sent on any edge in a round is bounded by a parameter b We consider two versions of the model: in the first, the players communicate by unicast, allowing them to send a different message on each of their links in one round; in the second, the players communicate by broadcast, sending one message to all their neighbors. It is known that the unicast version of the model is quite powerful; to date, no lower bounds for this model are known. In this paper we provide a partial explanation by showing that the unicast congested clique can simulate powerful classes of bounded-depth circuits, implying that even slightly super-constant lower bounds for the congested clique would give new lower bounds in circuit complexity. Moreover, under a widely-believed conjecture on matrix multiplication, the triangle detection problem, studied in [8], can be solved in O(ne) time for any e > 0. The broadcast version of the congested clique is the well-known multi-party shared-blackboard model of communication complexity (with number-in-hand input). This version is more amenable to lower bounds, and in this paper we show that the subgraph detection problem studied in [8] requires polynomially many rounds for several classes of subgraphs. We also give upper bounds for the subgraph detection problem, and relate the hardness of triangle detection in the broadcast congested clique to the communication complexity of set disjointness in the 3-party number-on-forehead model. Motivated by the need to understand the algorithmic foundations of distributed large-scale graph computations, we study some fundamental graph problems in a message-passing model for distributed computing where @math machines jointly perform computations on graphs with @math nodes (typically, @math ). The input graph is assumed to be initially randomly partitioned among the @math machines. Communication is point-to-point, and the goal is to minimize the number of communication rounds of the computation. We present (almost) tight bounds for the round complexity of two fundamental graph problems, namely PageRank computation and triangle enumeration. Our tight lower bounds, a main contribution of the paper, are established through an information-theoretic approach that relates the round complexity to the minimal amount of information required by machines for correctly solving a problem. Our approach is generic and might be useful in showing lower bounds in the context of similar problems and similar models. We show a lower bound of @math rounds for computing the PageRank. (Notation @math hides a @math factor.) We also present a simple distributed algorithm that computes the PageRank of all the nodes of a graph in @math rounds (notation @math hides a @math factor and an additive @math term). For triangle enumeration, we show a lower bound of @math rounds, where @math is the number of edges of the graph. Our result implies a lower bound of @math for the congested clique, which is tight up to logarithmic factors. We also present a distributed algorithm that enumerates all the triangles of a graph in @math rounds.