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. ChanEmail author
  • Konstantinos Tsakalidis


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).


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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