{"paper":{"title":"Removing Depth-Order Cycles Among Triangles: An Efficient Algorithm Generating Triangular Fragments","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Mark de Berg","submitted_at":"2017-01-03T12:41:18Z","abstract_excerpt":"More than 25 years ago Chazelle~\\emph{et al.} (FOCS 1991) studied the following question: Is it possible to cut any set of $n$ lines in ${\\Bbb R}^3$ into a subquadratic number of fragments such that the resulting fragments admit a depth order? They proved an $O(n^{9/4})$ bound for the very special case of bipartite weavings of lines. Since then only little progress was made, until a recent breakthrough by Aronov and Sharir (STOC 2016) who showed that $O(n^{3/2}\\mathrm{polylog}\\; n)$ fragments suffice for any set of lines. In a follow-up paper Aronov, Miller and Sharir (SODA 2017) proved an $O("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.00679","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"}