DISC 2012: Distributed Computing pp 195-209

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

(Extended Abstract)
• Danny Dolev
• Christoph Lenzen
• Shir Peled
Conference paper

DOI: 10.1007/978-3-642-33651-5_14

Volume 7611 of the book series Lecture Notes in Computer Science (LNCS)
Cite this paper as:
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

## Abstract

Let G = (V,E) be an n-vertex graph and Md a d-vertex graph, for some constant d. Is Md 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 Md 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.