{"paper":{"title":"Optimal Staged Self-Assembly of General Shapes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.ET"],"primary_cat":"cs.CG","authors_text":"Andrew Winslow, Cameron Chalk, Eric Martinez, Luis Vega, Robert Schweller, Tim Wylie","submitted_at":"2015-10-13T22:40:53Z","abstract_excerpt":"We analyze the number of tile types $t$, bins $b$, and stages necessary to assemble $n \\times n$ squares and scaled shapes in the staged tile assembly model. For $n \\times n$ squares, we prove $\\mathcal{O}(\\frac{\\log{n} - tb - t\\log t}{b^2} + \\frac{\\log \\log b}{\\log t})$ stages suffice and $\\Omega(\\frac{\\log{n} - tb - t\\log t}{b^2})$ are necessary for almost all $n$. For shapes $S$ with Kolmogorov complexity $K(S)$, we prove $\\mathcal{O}(\\frac{K(S) - tb - t\\log t}{b^2} + \\frac{\\log \\log b}{\\log t})$ stages suffice and $\\Omega(\\frac{K(S) - tb - t\\log t}{b^2})$ are necessary to assemble a scaled"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.03919","kind":"arxiv","version":3},"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"}