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

(Extended Abstract)
  • Danny Dolev
  • Christoph Lenzen
  • Shir Peled
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7611)


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alon, N.: Testing subgraphs in large graphs. Random Structures and Algorithms 21, 359–370 (2002)MathSciNetMATHCrossRefGoogle Scholar
  2. 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)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Chechik, S.: Message distribution technique (2011), private communicationGoogle Scholar
  4. 4.
    Chiba, N., Nishizeki, T.: Arboricity and subgraph listing algorithms. SIAM Journal on Computing 14, 210–223 (1985)MathSciNetMATHCrossRefGoogle Scholar
  5. 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. 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. 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)MathSciNetMATHCrossRefGoogle Scholar
  8. 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. 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)CrossRefGoogle Scholar
  10. 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. 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. 12.
    Lotker, Z., Patt-Shamir, B., Peleg, D.: Distributed MST for Constant Diameter Graphs. Distributed Computing 18(6) (2006)Google Scholar
  13. 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. 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. 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 CrossRefGoogle Scholar
  16. 16.
    Patt-Shamir, B., Teplitsky, M.: The Round Complexity of Distributed Sorting: Extended Abstract. In: PODC, pp. 249–256 (2011)Google Scholar
  17. 17.
    Peleg, D.: Distributed Computing: A Locality-Sensitive Approach. Society for Industrial and Applied Mathematics (2000)Google Scholar
  18. 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

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Danny Dolev
    • 1
  • Christoph Lenzen
    • 2
  • Shir Peled
    • 1
  1. 1.School of Engineering and Computer ScienceHebrew University of JerusalemIsrael
  2. 2.Department for Computer Science and Applied MathematicsWeizmann Institute of ScienceIsrael

Personalised recommendations