{"paper":{"title":"An Approximation Algorithm for Computing Shortest Paths in Weighted 3-d Domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS","cs.GR","cs.RO"],"primary_cat":"cs.CG","authors_text":"Anil Maheshwari, Hristo Djidjev, Joerg-Rudiger Sack, Lyudmil Aleksandrov","submitted_at":"2011-02-15T19:50:45Z","abstract_excerpt":"We present the first polynomial time approximation algorithm for computing shortest paths in weighted three-dimensional domains. Given a polyhedral domain $\\D$, consisting of $n$ tetrahedra with positive weights, and a real number $\\eps\\in(0,1)$, our algorithm constructs paths in $\\D$ from a fixed source vertex to all vertices of $\\D$, whose costs are at most $1+\\eps$ times the costs of (weighted) shortest paths, in $O(\\C(\\D)\\frac{n}{\\eps^{2.5}}\\log\\frac{n}{\\eps}\\log^3\\frac{1}{\\eps})$ time, where $\\C(\\D)$ is a geometric parameter related to the aspect ratios of tetrahedra. The efficiency of th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.3165","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"}