Optimal Deterministic Algorithms for 2-d and 3-d Shallow Cuttings

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 Matoušek (Comput Geom 2(3):169–186, 1992 ) and the optimal but randomized algor...

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Veröffentlicht in:Discrete & computational geometry Jg. 56; H. 4; S. 866 - 881
Hauptverfasser: Chan, Timothy M., Tsakalidis, Konstantinos
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 01.12.2016
Springer Nature B.V
Schlagworte:
Algorithms
Chips
Combinatorics
Computational geometry
Computational Mathematics and Numerical Analysis
Convexity
Hulls
Linear programming
Mathematics
Mathematics and Statistics
Optimization
Peeling
Planes
Searching
Halfspace range reporting
Shallow cuttings
Derandomization
Geometric data structures
ISSN:0179-5376, 1432-0444
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Zusammenfassung: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 Matoušek (Comput Geom 2(3):169–186, 1992 ) and the optimal but randomized algorithm of Ramos (Proceedings of the Fifteenth Annual Symposium on Computational Geometry, SoCG’99, 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, ε -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 Matoušek (Discrete Comput Geom 6(1):385–406, 1991 ) and Chazelle (Discrete Comput Geom 9(1):145–158, 1993 ).
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ISSN:0179-5376
1432-0444
DOI:10.1007/s00454-016-9784-4