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 |
|---|---|---|---|---|
| 2,997 | Subgraph Matching: on Compression and Computation | 2018 | VLDB | 7.7559339e-05 |
| 3,833 | Output-optimal Parallel Algorithms for Similarity Joins | 2017 | PODS | 6.7173578e-05 |
| 4,432 | Towards a Worst-Case I/O-Optimal Algorithm for Acyclic Joins | 2016 | PODS | 6.1938383e-05 |
| 4,708 | Instance and Output Optimal Parallel Algorithms for Acyclic Joins | 2019 | PODS | 5.980172e-05 |
| 4,779 | Join Dependency Testing, Loomis-Whitney Join, and Triangle Enumeration | 2015 | PODS | 5.9271735e-05 |
| 4,879 | Approximately Counting Triangles in Large Graph Streams Including Edge Duplicates with a Fixed Memory Usage | 2018 | VLDB | 5.8575676e-05 |
| 5,773 | I/O-Efficient Butterfly Counting at Scale | 2023 | SIGMOD | 5.3319911e-05 |
| 6,867 | 2-3 Cuckoo Filters for Faster Triangle Listing and Set Intersection | 2017 | PODS | 4.9025083e-05 |
| 8,589 | Output-Optimal Algorithms for Join-Aggregate Queries | 2025 | PODS | 4.4897014e-05 |
| 11,437 | Two-Attribute Skew Free, Isolated CP Theorem, and Massively Parallel Joins | 2021 | PODS | 4.1945683e-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 |
|---|---|---|---|---|
| 589 | Massive Graph Triangulation | 2013 | SIGMOD | 0.00019576567 |
| 1,308 | Upper and Lower Bounds on the Cost of a Map-Reduce Computation | 2013 | VLDB | 0.00012661651 |
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Semantically Similar Papers
| Overall Rank | Paper | Year | Venue | Pagerank |
|---|---|---|---|---|
| 9,228 | Efficiently Counting Triangles in Large Temporal Graphs | 2025 | SIGMOD | 4.3690661e-05 |
| 4,879 | Approximately Counting Triangles in Large Graph Streams Including Edge Duplicates with a Fixed Memory Usage | 2018 | VLDB | 5.8575676e-05 |
| 8,533 | How the Degeneracy Helps for Triangle Counting in Graph Streams | 2020 | PODS | 4.4937074e-05 |
| 1,344 | Counting and Sampling Triangles from a Graph Stream | 2013 | VLDB | 0.00012473724 |
| 6,867 | 2-3 Cuckoo Filters for Faster Triangle Listing and Set Intersection | 2017 | PODS | 4.9025083e-05 |
| 392 | Counting Triangles in Data Streams | 2006 | PODS | 0.00024556183 |
| 5,046 | Better Algorithms for Counting Triangles in Data Streams | 2016 | PODS | 5.7405307e-05 |
| 9,091 | Efficiently Enumerating Minimal Triangulations | 2017 | PODS | 4.39823e-05 |
| 8,540 | On Asymptotic Cost of Triangle Listing in Random Graphs | 2017 | PODS | 4.4937074e-05 |
| 589 | Massive Graph Triangulation | 2013 | SIGMOD | 0.00019576567 |