{"paper":{"title":"The Input/Output Complexity of Triangle Enumeration","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Francesco Silvestri, Rasmus Pagh","submitted_at":"2013-12-03T07:54:00Z","abstract_excerpt":"We consider the well-known problem of enumerating all triangles of an undirected graph. Our focus is on determining the input/output (I/O) complexity of this problem. Let $E$ be the number of edges, $M<E$ the size of internal memory, and $B$ the block size. The best results obtained previously are sort$(E^{3/2})$ I/Os (Dementiev, PhD thesis 2006) and $O(E^2/(MB))$ I/Os (Hu et al., SIGMOD 2013), where sort$(n)$ denotes the number of I/Os for sorting $n$ items. We improve the I/O complexity to $O(E^{3/2}/(\\sqrt{M} B))$ expected I/Os, which improves the previous bounds by a factor $\\min(\\sqrt{E/M"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.0723","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}