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

Authors Timothy M. Chan, Konstantinos Tsakalidis

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Timothy M. Chan
Konstantinos Tsakalidis

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Timothy M. Chan and Konstantinos Tsakalidis. Optimal Deterministic Algorithms for 2-d and 3-d Shallow Cuttings. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 719-732, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)


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).
  • shallow cuttings
  • derandomization
  • halfspace range reporting
  • geometric data structures


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