{"paper":{"title":"Finding small patterns in permutations in linear time","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"cs.DS","authors_text":"D\\'aniel Marx, Sylvain Guillemot","submitted_at":"2013-07-11T11:50:48Z","abstract_excerpt":"Given two permutations $\\sigma$ and $\\pi$, the \\textsc{Permutation Pattern} problem asks if $\\sigma$ is a subpattern of $\\pi$. We show that the problem can be solved in time $2^{O(\\ell^2\\log \\ell)}\\cdot n$, where $\\ell=|\\sigma|$ and $n=|\\pi|$. In other words, the problem is fixed-parameter tractable parameterized by the size of the subpattern to be found.\n  We introduce a novel type of decompositions for permutations and a corresponding width measure. We present a linear-time algorithm that either finds $\\sigma$ as a subpattern of $\\pi$, or finds a decomposition of $\\pi$ whose width is bounded"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.3073","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"}