{"paper":{"title":"A nearly optimal algorithm for deciding connectivity queries in smooth and bounded real algebraic sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.SC","authors_text":"Eric Schost, Mohab Safey El Din (LIP6, PolSys)","submitted_at":"2013-07-30T06:30:48Z","abstract_excerpt":"A roadmap for a semi-algebraic set $S$ is a curve which has a  non-empty and connected intersection with all connected components  of $S$. Hence, this kind of object, introduced by Canny, can be used  to answer connectivity queries (with applications, for instance, to  motion planning) but has also become of central importance in  effective real algebraic geometry, since it is used in higher-level  algorithms.  In this paper, we provide a probabilistic algorithm which computes  roadmaps for smooth and bounded real algebraic sets. Its output size  and running time are polynomial in $(nD)^{n\\log"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.7836","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"}