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“Tri, Tri Again”: Finding Triangles and Small Subgraphs in a Distributed Setting

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 7611)

Abstract

Let G = (V,E) be an n-vertex graph and M d a d-vertex graph, for some constant d. Is M d a subgraph of G? We consider this problem in a model where all n processes are connected to all other processes, and each message contains up to \(\mathcal{O}(\log n)\) bits. A simple deterministic algorithm that requires \(\mathcal{O}(n^{(d-2)/d}/\log n)\) communication rounds is presented. For the special case that M d is a triangle, we present a probabilistic algorithm that requires an expected \(\mathcal{O}(n^{1/3}/(t^ {2/3}+1))\) rounds of communication, where t is the number of triangles in the graph, and \(\mathcal{O}(\min\{n^{1/3}\log^{2/3}n/(t^ {2/3}+1),n^{1/3}\})\) with high probability.

We also present deterministic algorithms that are specially suited for sparse graphs. In graphs of maximum degree Δ, we can test for arbitrary subgraphs of diameter D in \(\mathcal{O}(\Delta^{D+1}/n)\) rounds. For triangles, we devise an algorithm featuring a round complexity of \(\mathcal{O}((A^2\log_{2+n/A^2} n)/n)\), where A denotes the arboricity of G.

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References

  1. Alon, N.: Testing subgraphs in large graphs. Random Structures and Algorithms 21, 359–370 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Alon, N., Kaufman, T., Krivelevich, M., Ron, D.: Testing triangle-freeness in general graphs. SIAM Journal on Discrete Math. 22(2), 786–819 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chechik, S.: Message distribution technique (2011), private communication

    Google Scholar 

  4. Chiba, N., Nishizeki, T.: Arboricity and subgraph listing algorithms. SIAM Journal on Computing 14, 210–223 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  5. Deo, N., Litow, B.: A Structural Approach to Graph Compression. In: Proc. 23rd International Symposium on Mathematical Foundations of Computer Science (MFCS), pp. 91–101 (1998)

    Google Scholar 

  6. Dolev, D., Lenzen, C., Peled, S.: ”Tri, Tri again”.: Finding Triangles and Small Subgraphs in a Distributed Setting. Computing Research Repository abs/1201.6652 (2012)

    Google Scholar 

  7. Gonen, M., Ron, D., Shavitt, Y.: Counting Stars and Other Small Subgraphs in Sublinear-Time. SIAM Journal on Discrete Mathematics 25(3), 1365–1411 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Grötzsch, H.: Zur Theorie der diskreten Gebilde, VII. Ein Dreifarbensatz für dreikreisfreie Netze auf der Kugel. In: Math.-Nat. Reihe., vol. 8, pp. 109–120. Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg (1958/1959)

    Google Scholar 

  9. Kashtan, N., Itzkovitz, S., Milo, R., Alon, U.: Efficient sampling algorithm for estimating subgraph concentrations and detecting network motifs. Bioinformatics 20(11), 1746–1758 (2004)

    Article  Google Scholar 

  10. Kothapalli, K., Scheideler, C., Onus, M., Schindelhauer, C.: Distributed Coloring in \(\tilde{\mathcal{O}}(\sqrt{\log n})\) Bit Rounds. In: IPDPS (2006)

    Google Scholar 

  11. Lenzen, C., Wattenhofer, R.: Tight Bounds for Parallel Randomized Load Balancing. In: Proc. 43rd Symposium on Theory of Computing (STOC), pp. 11–20 (2011)

    Google Scholar 

  12. Lotker, Z., Patt-Shamir, B., Peleg, D.: Distributed MST for Constant Diameter Graphs. Distributed Computing 18(6) (2006)

    Google Scholar 

  13. Lotker, Z., Pavlov, E., Patt-Shamir, B., Peleg, D.: MST Construction in \(\mathcal{O}(log log n)\) Communication Rounds. In: Proc. 15th Symposium on Parallel Algorithms and Architectures (SPAA), pp. 94–100 (2003)

    Google Scholar 

  14. McKay (mathoverflow.net/users/9025), B.: If many triangles share edges, then some edge is shared by many triangles. MathOverflow, http://mathoverflow.net/questions/83939 (version: 2011-12-20)

  15. Milo, R., Shen-Orr, S., Itzkovitz, S., Kashtan, N., Chklovskii, D., Alon, U.: Network Motifs: Simple Building Blocks of Complex Networks. Science 298(5594), 824–827 (2002), http://dx.doi.org/10.1126/science.298.5594.824

    Article  Google Scholar 

  16. Patt-Shamir, B., Teplitsky, M.: The Round Complexity of Distributed Sorting: Extended Abstract. In: PODC, pp. 249–256 (2011)

    Google Scholar 

  17. Peleg, D.: Distributed Computing: A Locality-Sensitive Approach. Society for Industrial and Applied Mathematics (2000)

    Google Scholar 

  18. Sarma, A.D., Holzer, S., Kor, L., Korman, A., Nanongkai, D., Pandurangan, G., Peleg, D., Wattenhofer, R.: Distributed Verification and Hardness of Distributed Approximation. In: 43rd Symposium on Theory of Computing, STOC (2011)

    Google Scholar 

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Dolev, D., Lenzen, C., Peled, S. (2012). “Tri, Tri Again”: Finding Triangles and Small Subgraphs in a Distributed Setting. In: Aguilera, M.K. (eds) Distributed Computing. DISC 2012. Lecture Notes in Computer Science, vol 7611. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-33651-5_14

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  • DOI: https://doi.org/10.1007/978-3-642-33651-5_14

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-33650-8

  • Online ISBN: 978-3-642-33651-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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