eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2015-06-12
719
732
10.4230/LIPIcs.SOCG.2015.719
article
Optimal Deterministic Algorithms for 2-d and 3-d Shallow Cuttings
Chan, Timothy M.
Tsakalidis, Konstantinos
We present optimal deterministic algorithms for constructing shallow cuttings in an arrangement of lines in two dimensions or planes in three dimensions. Our results improve the deterministic polynomial-time algorithm of Matousek (1992) and the optimal but randomized algorithm of Ramos (1999). This leads to efficient derandomization of previous algorithms for numerous well-studied problems in computational geometry, including halfspace range reporting in 2-d and 3-d, k nearest neighbors search in 2-d, (<= k)-levels in 3-d, order-k Voronoi diagrams in 2-d, linear programming with k violations in 2-d, dynamic convex hulls in 3-d, dynamic nearest neighbor search in 2-d, convex layers (onion peeling) in 3-d, epsilon-nets for halfspace ranges in 3-d, and more. As a side product we also describe an optimal deterministic algorithm for constructing standard (non-shallow) cuttings in two dimensions, which is arguably simpler than the known optimal algorithms by Matousek (1991) and Chazelle (1993).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol034-socg2015/LIPIcs.SOCG.2015.719/LIPIcs.SOCG.2015.719.pdf
shallow cuttings
derandomization
halfspace range reporting
geometric data structures