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
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In the standard CONGEST model for distributed network computing, it is known that "global" tasks such as minimum spanning tree, diameter, and all-pairs shortest paths, consume large bandwidth, for their running-time is @math rounds in @math -node networks with constant diameter. Surprisingly, "local" tasks such as detecting the presence of a 4-cycle as a subgraph also requires @math rounds, even using randomized algorithms, and the best known upper bound for detecting the presence of a 3-cycle is @math rounds. The objective of this paper is to better understand the landscape of such subgraph detection tasks. We show that, in contrast to , which are hard to detect in the CONGEST model, there exists a deterministic algorithm for detecting the presence of a subgraph isomorphic to @math running in a number of rounds, for every tree @math . Our algorithm provides a distributed implementation of a combinatorial technique due to Erd o for sparsening the set of partial solutions kept by the nodes at each round. Our result has important consequences to , i.e., to randomized algorithms whose aim is to distinguish between graphs satisfying a property, and graphs far from satisfying that property. In particular, we get that, for every graph pattern @math composed of an edge and a tree connected in an arbitrary manner, there exists a distributed testing algorithm for @math -freeness, performing in a constant number of rounds. Although the class of graph patterns @math formed by a tree and an edge connected arbitrarily may look artificial, all previous results of the literature concerning testing @math -freeness for classical patterns such as cycles and cliques can be viewed as direct consequences of our result, while our algorithm enables testing more complex patterns. 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. In the subgraph-freeness problem, we are given a constant-size graph @math , and wish to determine whether the network contains @math as a subgraph or not. The relaxation of the problem only requires us to distinguish graphs that are @math -free from graphs that are @math -far from @math -free, meaning an @math -fraction of their edges must be removed to obtain an @math -free graph. Recently, Censor-Hillel et. al. and showed that in the property-testing regime it is possible to test @math -freeness for any graph @math of size 4 in constant time, @math rounds, regardless of the network size. However, Fraigniaud et. al. also showed that their techniques for graphs @math of size 4 cannot test @math -cycle-freeness in constant time. In this paper we revisit the subgraph-freeness problem and show that @math -cycle-freeness, and indeed @math -freeness for many other graphs @math comprising more than 4 vertices, can be tested in constant time. We show that @math -freeness can be tested in @math rounds for any cycle @math , improving on the running time of @math of the previous algorithms for triangle-freeness and @math -freeness. In the special case of triangles, we show that triangle-freeness can be solved in @math rounds independently of @math , when @math is not too small with respect to the number of nodes and edges. We also show that @math -freeness for any constant-size tree @math can be tested in @math rounds, even without the property-testing relaxation. Building on these results, we define a general class of graphs for which we can test subgraph-freeness in @math rounds. This class includes all graphs over 5 vertices except the 5-clique, @math . For cliques @math over @math nodes, we show that @math -freeness can be tested in @math rounds, where @math is the number of edges. We introduce a new setting of algorithmic problems in random graphs, studying the minimum number of queries one needs to ask about the adjacency between pairs of vertices of G(n,p) in order to typically find a subgraph possessing a given target property. We show that if p≥lnn+lnlnn+ω(1)n, then one can find a Hamilton cycle with high probability after exposing (1+o(1))n edges. Our result is tight in both p and the number of exposed edges. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 2016