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)
Incoming Non-self Citations Over Time
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Incoming Citations (Sorted by Pagerank)
Showing 10 of 10 citing papers.
| Rank | Citing Paper | Year | Venue | Pagerank |
|---|---|---|---|---|
| 2,963 | Subgraph Matching: on Compression and Computation | 2018 | VLDB | 7.8061004e-05 |
| 3,831 | Output-optimal Parallel Algorithms for Similarity Joins | 2017 | PODS | 6.7146681e-05 |
| 4,430 | Towards a Worst-Case I/O-Optimal Algorithm for Acyclic Joins | 2016 | PODS | 6.1879319e-05 |
| 4,709 | Instance and Output Optimal Parallel Algorithms for Acyclic Joins | 2019 | PODS | 5.9744219e-05 |
| 4,783 | Join Dependency Testing, Loomis-Whitney Join, and Triangle Enumeration | 2015 | PODS | 5.9215677e-05 |
| 4,883 | Approximately Counting Triangles in Large Graph Streams Including Edge Duplicates with a Fixed Memory Usage | 2018 | VLDB | 5.8519327e-05 |
| 5,782 | I/O-Efficient Butterfly Counting at Scale | 2023 | SIGMOD | 5.3268721e-05 |
| 6,869 | 2-3 Cuckoo Filters for Faster Triangle Listing and Set Intersection | 2017 | PODS | 4.8978627e-05 |
| 8,587 | Output-Optimal Algorithms for Join-Aggregate Queries | 2025 | PODS | 4.4853975e-05 |
| 11,440 | Two-Attribute Skew Free, Isolated CP Theorem, and Massively Parallel Joins | 2021 | PODS | 4.1905499e-05 |
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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 |
|---|---|---|---|---|
| 588 | Massive Graph Triangulation | 2013 | SIGMOD | 0.00019588834 |
| 1,312 | Upper and Lower Bounds on the Cost of a Map-Reduce Computation | 2013 | VLDB | 0.00012650678 |
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Semantically Similar Papers
| Overall Rank | Paper | Year | Venue | Pagerank |
|---|---|---|---|---|
| 9,231 | Efficiently Counting Triangles in Large Temporal Graphs | 2025 | SIGMOD | 4.3648789e-05 |
| 4,883 | Approximately Counting Triangles in Large Graph Streams Including Edge Duplicates with a Fixed Memory Usage | 2018 | VLDB | 5.8519327e-05 |
| 8,532 | How the Degeneracy Helps for Triangle Counting in Graph Streams | 2020 | PODS | 4.4893996e-05 |
| 1,348 | Counting and Sampling Triangles from a Graph Stream | 2013 | VLDB | 0.00012461666 |
| 6,869 | 2-3 Cuckoo Filters for Faster Triangle Listing and Set Intersection | 2017 | PODS | 4.8978627e-05 |
| 389 | Counting Triangles in Data Streams | 2006 | PODS | 0.00024649634 |
| 5,043 | Better Algorithms for Counting Triangles in Data Streams | 2016 | PODS | 5.7350154e-05 |
| 9,088 | Efficiently Enumerating Minimal Triangulations | 2017 | PODS | 4.3940132e-05 |
| 8,538 | On Asymptotic Cost of Triangle Listing in Random Graphs | 2017 | PODS | 4.4893996e-05 |
| 588 | Massive Graph Triangulation | 2013 | SIGMOD | 0.00019588834 |