This paper provides an in-depth study of the fundamental problems of finding small subgraphs in distributed dynamic networks. While some problems are trivially easy to handle, such as detecting a triangle that emerges after an edge insertion, we show that, perhaps somewhat surprisingly, other problems exhibit a wide range of complexities in terms of the trade-offs between their round and bandwidth complexities. In the case of triangles, which are only affected by the topology of the immediate neighborhood, some end results are: The bandwidth complexity of @math -round dynamic triangle detection or listing is @math . The bandwidth complexity of @math -round dynamic triangle membership listing is @math for node edge deletions, @math for edge insertions, and @math for node insertions. The bandwidth complexity of @math -round dynamic triangle membership detection is @math for node edge deletions, @math for edge insertions, and @math for node insertions. Most of our upper and lower bounds are . Additionally, we provide almost always tight upper and lower bounds for larger cliques.
A related problem is property-testing of subgraphs, where the nodes need to determine, for a given subgraph @math , whether the graph is @math -free or from being @math -free @cite_10 @cite_18 @cite_17 @cite_5 @cite_14 @cite_16 .
@cite_18 @cite_14 @cite_5 @cite_16 @cite_10 @cite_17
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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. In this paper we present distributed property-testing algorithms for graph properties in the CONGEST model, with emphasis on testing subgraph-freeness. Testing a graph property P means distinguishing graphs G = (V,E) having property P from graphs that are epsilon-far from having it, meaning that epsilon|E| edges must be added or removed from G to obtain a graph satisfying P. We present a series of results, including: - Testing H-freeness in O(1 epsilon) rounds, for any constant-sized graph H containing an edge (u,v) such that any cycle in H contain either u or v (or both). This includes all connected graphs over five vertices except K_5. For triangles, we can do even better when epsilon is not too small. - A deterministic CONGEST protocol determining whether a graph contains a given tree as a subgraph in constant time. - For cliques K_s with s >= 5, we show that K_s-freeness can be tested in O(m^(1 2-1 (s-2)) epsilon^(-1 2-1 (s-2))) rounds, where m is the number of edges in the network graph. - We describe a general procedure for converting epsilon-testers with f(D) rounds, where D denotes the diameter of the graph, to work in O((log n) epsilon)+f((log n) epsilon) rounds, where n is the number of processors of the network. We then apply this procedure to obtain an epsilon-tester for testing whether a graph is bipartite and testing whether a graph is cycle-free. Moreover, for cycle-freeness, we obtain a corrector of the graph that locally corrects the graph so that the corrected graph is acyclic. Note that, unlike a tester, a corrector needs to mend the graph in many places in the case that the graph is far from having the property. These protocols extend and improve previous results of [Censor- 2016] and [ 2016]. We consider distribution-free property-testing of graph connectivity. In this setting of property testing, the distance between functions is measured with respect to a fixed but unknown distribution D on the domain, and the testing algorithm has an oracle access to random sampling from the domain according to this distribution D. This notion of distribution-free testing was previously defined, and testers were shown for very few properties. However, no distribution-free property testing algorithm was known for any graph property. We present the first distribution-free testing algorithms for one of the central properties in this area—graph connectivity (specifically, the problem is mainly interesting in the case of sparse graphs). We introduce three testing models for sparse graphs: A model for bounded-degree graphs, A model for graphs with a bound on the total number of edges (both models were already considered in the context of uniform distribution testing), and A model which is a combination of the two previous testing models; i.e., bounded-degree graphs with a bound on the total number of edges. We prove that connectivity can be tested in each of these testing models, in a distribution-free manner, using a number of queries that is independent of the size of the graph. This is done by providing a new analysis to previously known connectivity testers (from “standard”, uniform distribution property-testing) and by introducing some new testers. We consider the following class of problems: given a graph, find the maximum number of nodes inducing a subgraph that satisfies a desired property π, such as planar, acyclic, bipartite, etc. We show that this problem is hard to approximate for any property π on directed or undirected graphs that is nontrivial and hereditary on induced subgraphs.