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Discrete & Computational Geometry

, Volume 56, Issue 4, pp 866–881 | Cite as

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

  • Timothy M. Chan
  • Konstantinos Tsakalidis
Article

Abstract

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, \(({\le }k)\)-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, \(\varepsilon \)-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).

Keywords

Shallow cuttings Derandomization Halfspace range reporting Geometric data structures 

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Computer Engineering and Informatics DepartmentUniversity of PatrasPatrasGreece

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