The Input/Output Complexity of Triangle Enumeration

Summary: Randomized cache-oblivious algorithm enumerates all triangles in an undirected graph in O(E^{3/2}/√(M B)) expected I/Os (deterministic cache-aware variant when M ≥ E^ε), improving prior bounds. Proves tight I/O lower bound Ω(t/√(M B)) (worst-case t = Θ(E^{3/2})), and introduces a new color-coding technique for I/O-efficient listing. (summarized by gpt-5-mini on Feb 09 2026)

Paper ID: h17bca48e60b1a99b
Venue: PODS
Year: 2014
Pagerank: 8.7890942e-05
Overall Rank: 2,229 | 85.07%
DOI: 10.1145/2594538.2594552

Incoming Non-self Citations Over Time

Authors

1. Rasmus Pagh (IT University of Copenhagen)
2. Francesco Silvestri (IT University of Copenhagen; University of Padova)

BibTeX Citation

@inproceedings{pagh_pods14,
        address = {New York, NY, USA},
        series = {{PODS} '14},
        title = {{The Input/Output Complexity of Triangle Enumeration}},
        url = {https://dl.acm.org/doi/10.1145/2594538.2594552},
        doi = {10.1145/2594538.2594552},
        booktitle = {Proceedings of the {ACM} {SIGMOD} Symposium on {Principles} of {Database} {Systems}},
        publisher = {Association for Computing Machinery},
        author = {Pagh, Rasmus and Silvestri, Francesco},
        year = {2014}
}

Incoming Citations (Sorted by Pagerank)

Showing 10 of 10 citing papers.

Rank	Citing Paper	Year	Venue	Pagerank
2,128	Subgraph Matching: on Compression and Computation	2018	VLDB	8.9879343e-05
2,475	Output-optimal Parallel Algorithms for Similarity Joins	2017	PODS	8.3947133e-05
4,487	Instance and Output Optimal Parallel Algorithms for Acyclic Joins	2019	PODS	6.5628751e-05
4,497	Towards a Worst-Case I/O-Optimal Algorithm for Acyclic Joins	2016	PODS	6.5592387e-05
4,511	Approximately Counting Triangles in Large Graph Streams Including Edge Duplicates with a Fixed Memory Usage	2018	VLDB	6.5538597e-05
5,093	Join Dependency Testing, Loomis-Whitney Join, and Triangle Enumeration	2015	PODS	6.2653284e-05
5,576	I/O-Efficient Butterfly Counting at Scale	2023	SIGMOD	6.062102e-05
6,585	Output-Optimal Algorithms for Join-Aggregate Queries	2025	PODS	5.7238683e-05
6,980	2-3 Cuckoo Filters for Faster Triangle Listing and Set Intersection	2017	PODS	5.6128504e-05
11,992	Two-Attribute Skew Free, Isolated CP Theorem, and Massively Parallel Joins	2021	PODS	4.9605343e-05

Outgoing Citations (Sorted by Pagerank)

Showing 2 of 2 cited papers.

Citations counted here include only citations to other VLDB/SIGMOD/CIDR/PODS papers in this database.

Rank	Cited Paper	Year	Venue	Pagerank
599	Massive Graph Triangulation	2013	SIGMOD	0.00015700433
1,547	Upper and Lower Bounds on the Cost of a Map-Reduce Computation	2013	VLDB	0.00010268377

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1	8,670	Efficiently Counting Triangles in Large Temporal Graphs	2025	SIGMOD
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3	9,118	How the Degeneracy Helps for Triangle Counting in Graph Streams	2020	PODS
4	1,125	Counting and Sampling Triangles from a Graph Stream	2013	VLDB
5	6,980	2-3 Cuckoo Filters for Faster Triangle Listing and Set Intersection	2017	PODS
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