Advertisement

“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)

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.

Keywords

Deterministic Algorithm Network Motif Sparse Graph Neighbor List Communication Round 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

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