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.
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Björklund, A., Pagh, R., Williams, V.V., Zwick, U. (2014). Listing Triangles. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds) Automata, Languages, and Programming. ICALP 2014. Lecture Notes in Computer Science, vol 8572. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-43948-7_19
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DOI: https://doi.org/10.1007/978-3-662-43948-7_19
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