Abstract
We study property testing in the context of distributed computing, under the classical CONGEST model. It is known that testing whether a graph is triangle-free can be done in a constant number of rounds, where the constant depends on how far the input graph is from being triangle-free. We show that, for every connected 4-node graph H, testing whether a graph is H-free can be done in a constant number of rounds too. The constant also depends on how far the input graph is from being H-free, and the dependence is identical to the one in the case of testing triangle-freeness. Hence, in particular, testing whether a graph is \(K_4\)-free, and testing whether a graph is \(C_4\)-free can be done in a constant number of rounds (where \(K_k\) denotes the k-node clique, and \(C_k\) denotes the k-node cycle). On the other hand, we show that testing \(K_k\)-freeness and \(C_k\)-freeness for \(k\ge 5\) appear to be much harder. Specifically, we investigate two natural types of generic algorithms for testing H-freeness, called DFS tester and BFS tester. The latter captures the previously known algorithm to test the presence of triangles, while the former captures our generic algorithm to test the presence of a 4-node graph pattern H. We prove that both DFS and BFS testers fail to test \(K_k\)-freeness and \(C_k\)-freeness in a constant number of rounds for \(k\ge 5\).
Additional support from ANR project DISPLEXITY, Inria project GANG, CONICYT via Basal in Applied Mathematics, Núcleo Milenio Información y Coordinación en Redes ICM/FIC RC130003, Fondecyt 1130061 and Fondecyt 3150552.
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Notes
- 1.
Actually, property testing tackles graph problems in both the dense model (graphs represented by adjacency matrices) and the sparse model (graphs represented by adjacency lists). In this paper, we are interested in property testing in the sparse model.
- 2.
The interested reader can consult [29] for the state-of-the-art on such combinatorial constructions, in particular constructions for \(p' \ge p^{1 - c /\sqrt{\log p}}\), for a constant c depending on k.
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Fraigniaud, P., Rapaport, I., Salo, V., Todinca, I. (2016). Distributed Testing of Excluded Subgraphs. In: Gavoille, C., Ilcinkas, D. (eds) Distributed Computing. DISC 2016. Lecture Notes in Computer Science(), vol 9888. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53426-7_25
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