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.
References
Alon, N., Kaufman, T., Krivelevich, M., Ron, D.: Testing triangle-freeness in general graphs. SIAM J. Discrete Math. 22(2), 786–819 (2008)
Arfaoui, H., Fraigniaud, P., Ilcinkas, D., Mathieu, F.: Distributedly testing cycle-freeness. In: Kratsch, D., Todinca, I. (eds.) WG 2014. LNCS, vol. 8747, pp. 15–28. Springer, Heidelberg (2014)
Arfaoui, H., Fraigniaud, P., Pelc, A.: Local decision and verification with bounded-size outputs. In: Higashino, T., Katayama, Y., Masuzawa, T., Potop-Butucaru, M., Yamashita, M. (eds.) SSS 2013. LNCS, vol. 8255, pp. 133–147. Springer, Heidelberg (2013)
Becker, F., Kosowski, A., Matamala, M., Nisse, N., Rapaport, I., Suchan, K., Todinca, I.: Allowing each node to communicate only once in a distributed system: shared whiteboard models. Distrib. Comput. 28(3), 189–200 (2015)
Behrend, F.A.: On sets of integers which contain no three terms in arithmetical progression. Proc. Natl. Acad. Sci. 32(12), 331 (1946)
Censor-Hillel, K., Fischer, E., Schwartzman, G., Vasudev, Y.: Fast distributed algorithms for testing graph properties. CoRR abs/1602.03718, February 2016
Censor-Hillel, K., Kaski, P., Korhonen, J.H., Lenzen, C., Paz, A., Suomela, J.: Algebraic methods in the congested clique. In: Proceedings of PODC 2015, pp. 143–152 (2015)
Das-Sarma, A., Holzer, S., Kor, L., Korman, A., Nanongkai, D., Pandurangan, G., Peleg, D., Wattenhofer, R.: Distributed verification and hardness of distributed approximation. SIAM J. Comput. 41(5), 1235–1265 (2012)
Dolev, D., Lenzen, C., Peled, S.: “Tri, Tri Again”: finding triangles and small subgraphs in a distributed setting. In: Aguilera, M.K. (ed.) DISC 2012. LNCS, vol. 7611, pp. 195–209. Springer, Heidelberg (2012)
Drucker, A., Kuhn, F., Oshman, R.: On the power of the congested clique model. In: Proceedings of PODC 2014, pp. 367–376 (2014)
Elkin, M.: An unconditional lower bound on the time-approximation trade-off for the distributed minimum spanning tree problem. SIAM J. Comput. 36(2), 433–456 (2006)
Fraigniaud, P., Göös, M., Korman, A., Suomela, J.: What can be decided locally without identifiers? In: Proceedings of PODC 2013, pp. 157–165 (2013)
Fraigniaud, P., Korman, A., Peleg, D.: Local distributed decision. In: Proceedings of FOCS 2011, pp. 708–717 (2011)
Fraigniaud, P., Rajsbaum, S., Travers, C.: Locality and checkability in wait-free computing. Distrib. Comput. 26(4), 223–242 (2013)
Fraigniaud, P., Rajsbaum, S., Travers, C.: On the number of opinions needed for fault-tolerant run-time monitoring in distributed systems. In: Bonakdarpour, B., Smolka, S.A. (eds.) RV 2014. LNCS, vol. 8734, pp. 92–107. Springer, Heidelberg (2014)
Goldreich, O. (ed.): Property Testing. LNCS, vol. 6390. Springer, Heidelberg (2010)
Goldreich, O., Goldwasser, S., Ron, D.: Property testing and its connection to learning and approximation. J. ACM 45(4), 653–750 (1998)
Göös, M., Jukka Suomela, J.: Locally checkable proofs. In: Proceedings of PODC 2011, pp. 159–168 (2011)
Hegeman, J.W., Pandurangan, G., Pemmaraju, S.V., Sardeshmukh, V.B., Scquizzato, M.: Toward optimal bounds in the congested clique: graph connectivity and MST. In: Proceedings of PODC 2015, pp. 91–100 (2015)
Hegeman, J.W., Pemmaraju, S.V., Sardeshmukh, V.B.: Near-constant-time distributed algorithms on a congested clique. In: Kuhn, F. (ed.) DISC 2014. LNCS, vol. 8784, pp. 514–530. Springer, Heidelberg (2014)
Kari, J., Matamala, M., Rapaport, I., Salo, V.: Solving the induced subgraphproblem in the randomized multiparty simultaneous messages model. In: Scheideler, C. (ed.) SIROCCO 2015. LNCS, vol. 9439, pp. 370–384. Springer, Heidelberg (2015)
Lenzen, C.: Optimal deterministic routing and sorting on the congested clique. In: Proceedings of PODC 2013, pp. 42–50 (2013)
Lotker, Z., Patt-Shamir, B., Peleg, D.: Distributed MST for constant diameter graphs. Distrib. Comput. 18(6), 453–460 (2006)
Lotker, Z., Pavlov, E., Patt-Shamir, B., Peleg, D.: MST construction in \({{\cal O}}(\log \log n)\) communication rounds. In: Proceedings of SPAA 2003, pp. 94–100 (2003)
Mitzenmacher, M., Upfal, E.: Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press, Cambridge (2005)
Patt-Shamir, B., Teplitsky, M.: The round complexity of distributed sorting. In: Proceedings of PODC 2011, pp. 249–256 (2011)
Peleg, D.: Distributed Computing: A Locality-sensitive Approach. SIAM, Philadelphia (2000)
Reiter, F.: Distributed graph automata. In: Proceedings of LICS 2015, pp. 192–201 (2015)
Schoen, T., Shkredov, I.D.: Roth’s theorem in many variables. Isr. J. Math. 199(1), 287–308 (2014)
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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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