Listing triangles

Andreas Björklund, Rasmus Pagh, Virginia Vassilevska Williams, Uri Zwick

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


We present new algorithms for listing triangles in dense and sparse graphs. The running time of our algorithm for dense graphs is Õ(n ω + n3(ω-1)/(5-ω)t 2(3-ω)/(5-ω)), and the running time of the algorithm for sparse graphs is Õ(m2ω/(ω+1) + m 3(ω-1)/(ω+1)t(3-ω)/(ω+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 Õ(n2.373 + n1.568 t0.478) and Õ(m1.408 + m1.222 t0.186), 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 Õ(n2 + nt 2/3) and Õ(m4/3 + mt1/3), respectively. In particular, if ω = 2, our algorithm lists m triangles in Õ(m 4/3) time. Pǎtraşcu (STOC 2010) showed that Ω(m4/3-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.

Original languageEnglish
Title of host publicationAutomata, Languages, and Programming - 41st International Colloquium, ICALP 2014, Proceedings
PublisherSpringer Verlag
Number of pages12
EditionPART 1
ISBN (Print)9783662439470
StatePublished - 2014
Event41st International Colloquium on Automata, Languages, and Programming, ICALP 2014 - Copenhagen, Denmark
Duration: 8 Jul 201411 Jul 2014

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
NumberPART 1
Volume8572 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference41st International Colloquium on Automata, Languages, and Programming, ICALP 2014


FundersFunder number
Bloom's Syndrome FoundationBSF:2012338
National Stroke FoundationCCF-1417238


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