ICALP 2014: Automata, Languages, and Programming pp 223-234

# Listing Triangles

• Andreas Björklund
• Rasmus Pagh
• Virginia Vassilevska Williams
• Uri Zwick
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8572)

## Abstract

We present new algorithms for listing triangles in dense and sparse graphs. The running time of our algorithm for dense graphs is $$\tilde{\mathcal O}(n^{\omega} + n^{3(\omega-1)/(5-\omega)} t^{2(3-\omega)/(5-\omega)})$$, and the running time of the algorithm for sparse graphs is $$\tilde{\mathcal O}(m^{{2\omega}/{(\omega+1)}} + m^{{3(\omega-1)}/{(\omega+1)}} t^{{(3-\omega)}/{(\omega+1)}})$$, where n is the number of vertices, m is the number of edges, t is the number of triangles to be listed, and ω < 2.373 is the exponent of fast matrix multiplication. With the current bound on ω, the running times of our algorithms are $$\tilde{\mathcal O}( n^{2.373} + n^{1.568}\, t^{0.478})$$ and $$\tilde{\mathcal O}(m^{1.408} + m^{1.222}\, t^{0.186})$$, respectively. We first obtain randomized algorithms with the desired running times and then derandomize them using sparse recovery techniques.

If ω = 2, the running times of the algorithms become $$\tilde{\mathcal O}(n^2+nt^{2/3})$$ and $$\tilde{\mathcal O}(m^{4/3}+mt^{1/3})$$, respectively. In particular, if ω = 2, our algorithm lists m triangles in $$\tilde{\mathcal O}(m^{4/3})$$ time. Pǎtraşcu (STOC 2010) showed that Ω(m 4/3 − o(1)) time is required for listing m triangles, unless there exist subquadratic algorithms for 3SUM. We show that unless one can solve quadratic equation systems over a finite field significantly faster than the brute force algorithm, our triangle listing runtime bounds are tight assuming ω = 2, also for graphs with more triangles.

## Keywords

Sparse Graph Quadratic Equation System Sparse Recovery High Degree Vertex Sparse Signal Recovery
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.

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## Authors and Affiliations

• Andreas Björklund
• 1
• Rasmus Pagh
• 2
• Virginia Vassilevska Williams
• 3
• Uri Zwick
• 4
1. 1.Department of Computer ScienceLund UniversitySweden
2. 2.IT University of CopenhagenDenmark
3. 3.Computer Science DepartmentStanford UniversityUSA
4. 4.Blavatnik School of Computer ScienceTel Aviv UniversityIsrael