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.
As noted before, @cite_24 have studied lower bounds for cycle detection in the model. Specifically, their lower bound for @math -cycle detection is @math rounds, where @math is the n number for cycles, that is, the maximum number of edges in a @math -cycle-free graph with @math nodes. This lower bound is @math for odd cycles, and @math for @math -cycles. However, for longer even cycles, this can give at most @math due to known bounds for Tur ' a n numbers, and matching bounds on Tur ' a n numbers are only known for @math and @math ; see e.g. Pikhurko @cite_8 and references therein.
@cite_24 @cite_8
2107805020 2019898392
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. The Tur´an function ex(n, F) is the maximum number of edges in an F-free graph on n vertices. The question of estimating this function for F = C2k, the cycle of length 2k, is one of the central open questions in this area that goes back to the 1930s. We prove that ex(n,C2k) ≤ (k − 1) n1+1 k + 16(k − 1)n, improving the previously best known general upper bound of Verstra¨ete [Combin. Probab. Computing 9 (2000), 369–373] by a factor 8 + o(1) when n � k.