The congested clique model of distributed computing has been receiving attention as a model for densely connected distributed systems. While there has been significant progress on the side of upper bounds, we have very little in terms of lower bounds for the congested clique; indeed, it is now known that proving explicit congested clique lower bounds is as difficult as proving circuit lower bounds. In this work, we use various more traditional complexity theory tools to build a clearer picture of the complexity landscape of the congested clique: item Nondeterminism and beyond: We introduce the nondeterministic congested clique model (analogous to NP) and show that there is a natural canonical problem family that captures all problems solvable in constant time with nondeterministic algorithms. We further generalise these notions by introducing the constant-round decision hierarchy (analogous to the polynomial hierarchy). item Non-constructive lower bounds: We lift the prior non-uniform counting arguments to a general technique for proving non-constructive uniform lower bounds for the congested clique. In particular, we prove a time hierarchy theorem for the congested clique, showing that there are decision problems of essentially all complexities, both in the deterministic and nondeterministic settings. item Fine-grained complexity: We map out relationships between various natural problems in the congested clique model, arguing that a reduction-based complexity theory currently gives us a fairly good picture of the complexity landscape of the congested clique.
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. We consider a message passing model with n nodes, each connected to all other nodes by a link that can deliver a message of B bits in a time unit (typically, B = O(log n)). We assume that each node has an input of size L bits (typically, L = O(n log n)) and the nodes cooperate in order to compute some function (i.e., perform a distributed task). We are interested in the number of rounds required to compute the function. We give two results regarding this model. First, we show that most boolean functions require ‸ L B ‹ − 1 rounds to compute deterministically, and that even if we consider randomized protocols that are allowed to err, the expected running time remains Ω(L B) for most boolean function. Second, trying to find explicit functions that require superconstant time, we consider the pointer chasing problem. In this problem, each node i is given an array Ai of length n whose entries are in [n], and the task is to find, for any j ∈ [n], the value of An-1[An-2[. . .A0[j] . . .]]. We give a deterministi...