{"paper":{"title":"Fourier meets M\\\"{o}bius: fast subset convolution","license":"","headline":"","cross_cats":["cs.DM","math.CO"],"primary_cat":"cs.DS","authors_text":"Andreas Bj\\\"orklund, Mikko Koivisto, Petteri Kaski, Thore Husfeldt","submitted_at":"2006-11-21T08:34:30Z","abstract_excerpt":"We present a fast algorithm for the subset convolution problem: given functions f and g defined on the lattice of subsets of an n-element set N, compute their subset convolution f*g, defined for all S\\subseteq N by (f * g)(S) = \\sum_{T \\subseteq S}f(T) g(S\\setminus T), where addition and multiplication is carried out in an arbitrary ring. Via M\\\"{o}bius transform and inversion, our algorithm evaluates the subset convolution in O(n^2 2^n) additions and multiplications, substantially improving upon the straightforward O(3^n) algorithm. Specifically, if the input functions have an integer range {"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"cs/0611101","kind":"arxiv","version":1},"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"}