{"paper":{"title":"Almost Tight Bounds for Eliminating Depth Cycles in Three Dimensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Boris Aronov, Micha Sharir","submitted_at":"2015-12-01T17:43:00Z","abstract_excerpt":"Given $n$ non-vertical lines in 3-space, their vertical depth (above/below) relation can contain cycles. We show that the lines can be cut into $O(n^{3/2}\\mathop{\\mathrm{polylog}} n)$ pieces, such that the depth relation among these pieces is now a proper partial order. This bound is nearly tight in the worst case.\n  Previous results on this topic could only handle restricted cases of the problem (such as handling only triangular cycles, by Aronov, Koltun, and Sharir (2005), or only cycles in grid-like patterns, by Chazelle et al. (1992)), and the bounds were considerably weaker---much closer "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.00358","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"}