Three Notes on Distributed Property Testing

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 ε-far from having it, meaning that ε|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/ε) 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 K5. For triangles, we can do even better when ε is not too small. A deterministic congest protocol determining whether a graph contains a given tree as a subgraph in constant time. For cliques Ks with s ≥ 5, we show that Ks-freeness can be tested in O(m 1 2− 1 s−2 · ε− 1 2− 1 s−2 ) rounds, where m is the number of edges in the network graph. We describe a general procedure for converting ε-testers with f(D) rounds, where D denotes the diameter of the graph, to work in O((logn)/ε) + f((logn)/ε) rounds, where n is the number of processors of the network. We then apply this procedure to obtain an ε-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. ∗ Work done while visiting Max Planck Institute for Informatics. † Additional support from ANR Project DESCARTES, and from INRIA Project GANG. ‡ This work was partially supported by CONICYT via Basal in Applied Mathematics § Orr Fischer, Tzlil Gonen and Rotem Oshman are supported by the Israeli Centers of Research Excellence (I-CORE) program, (Center No.4/11) and by BSF Grant No. 2014256. ¶ This work was partially supported by Fondecyt 1170021, Núcleo Milenio Información y Coordinación en Redes ICM/FIC RC130003 © Guy Even, Orr Fischer, Pierre Fraigniaud , Tzlil Gonen, Reut Levi, Moti Medina, Pedro Montealegre, Dennis Olivetti, Rotem Oshman, Ivan Rapaport and Ioan Todinca; licensed under Creative Commons License CC-BY 31st International Symposium on Distributed Computing (DISC 2017). Editor: Andréa Richa; Article No. 15; pp. 15:1–15:30 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 15:2 Three Notes on Distributed Property Testing These protocols extend and improve previous results of [Censor-Hillel et al. 2016] and [Fraigniaud et al. 2016]. 1998 ACM Subject Classification F.2 ANALYSIS OF ALGORITHMS AND PROBLEM COMPLEXITY Digital Object Identifier 10.4230/LIPIcs.DISC.2017.15 1 General Introduction Distributed decision refers to tasks in which the computing elements of a distributed system have to collectively decide whether the system satisfies some given boolean predicate on system states. If the system state is legal, i.e., it satisfies the given predicate, then all computing elements must accept; if the system state is illegal, then at least one computing element must reject. Distributed property testing is a relaxed variant of distributed decision, which only requires distinguishing legal states from states that are “far from” being legal. (The notion of “farness” depends on the context.) In the context of distributed network computing, one is interested in deciding or testing whether the actual network, modeled as a simple connected graph, satisfies some given predicate on graphs; e.g., bipartiteness, cycle-freeness, subgraph-freeness, etc. For a positive distance parameter ε ≤ 1, a graph G with m edges is said to be ε-far from satisfying a given property P if removing and/or adding up to εm edges from/to G cannot result in a graph satisfying P . In this paper we study distributed decision in general, and distributed property testing in particular, in the framework of distributed network computing, under the standard congest model. This paper is the result of merging the three papers [16], [17], and [18] that were concurrently submitted to the 31st International Symposium on Distributed Computing (DISC 2017), which independently showed overlapping results, using different methods and ideas. To highlight the different approaches to the problem, we chose to present a short version of each of the three papers in the form of three notes. The Subgraph-Freeness Problem. Each of the three notes presented here gives results on subgraph-freeness: we are given a constant-size graph H, and we wish to determine whether the network graph contains H as a subgraph or not. In the property testing relaxation of the problem, we only need to distinguish the case where the network graph is H-free from the case where it is ε-far from H-free, in the sense that at least an ε-fraction of the graph’s edges must be removed to eliminate all copies of H. Hereafter, we provide a summary of the results and methods in each paper. 1.1 Summary of the Results and Techniques Note #1: Color-Coding Based Algorithms for Testing Subgraph-Freeness. This note uses a technique called color-coding [4] to design randomized algorithms for property-testing subgraph-freeness in O(1/ε) rounds, for any subgraph H that contains an edge (u, v) such that any cycle in H contains at least one of u and v. In the case of trees, the color-coding technique yields an O(1)-round algorithm for testing exactly whether the graph contains the given tree or not. In addition, for cliques Ks with s ≥ 3, we show that Ks-freeness can be tested in O(m 1 2− 1 s−2 ·ε− 1 2− 1 s−2 ) rounds, wherem is the number of edges in the network graph.