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: 8.8194236e-05
Overall Rank: 2,215 | 85.11%
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
2,166	Subgraph Matching: on Compression and Computation	2018	VLDB	8.9334874e-05
2,464	Output-optimal Parallel Algorithms for Similarity Joins	2017	PODS	8.4260608e-05
4,463	Instance and Output Optimal Parallel Algorithms for Acyclic Joins	2019	PODS	6.5877664e-05
4,475	Towards a Worst-Case I/O-Optimal Algorithm for Acyclic Joins	2016	PODS	6.5838041e-05
4,487	Approximately Counting Triangles in Large Graph Streams Including Edge Duplicates with a Fixed Memory Usage	2018	VLDB	6.5786974e-05
5,069	Join Dependency Testing, Loomis-Whitney Join, and Triangle Enumeration	2015	PODS	6.2890683e-05
5,558	I/O-Efficient Butterfly Counting at Scale	2023	SIGMOD	6.0850942e-05
6,573	Output-Optimal Algorithms for Join-Aggregate Queries	2025	PODS	5.7455776e-05
6,957	2-3 Cuckoo Filters for Faster Triangle Listing and Set Intersection	2017	PODS	5.6341313e-05
11,942	Two-Attribute Skew Free, Isolated CP Theorem, and Massively Parallel Joins	2021	PODS	4.9793485e-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
600	Massive Graph Triangulation	2013	SIGMOD	0.00015740352
1,545	Upper and Lower Bounds on the Cost of a Map-Reduce Computation	2013	VLDB	0.00010302384

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1	8,645	Efficiently Counting Triangles in Large Temporal Graphs	2025	SIGMOD
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3	9,089	How the Degeneracy Helps for Triangle Counting in Graph Streams	2020	PODS
4	1,121	Counting and Sampling Triangles from a Graph Stream	2013	VLDB
5	6,957	2-3 Cuckoo Filters for Faster Triangle Listing and Set Intersection	2017	PODS
6	462	Counting Triangles in Data Streams	2006	PODS
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