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Shallow cuttings are one of the most fundamental tools in range searching as many problems in the field admit efficient static data structures due to their application. We present the first efficient deterministic algorithms that, given n three-dimensional points, construct optimal-size (single and multiple) shallow cuttings for orthogonal dominance ranges. Specifically, we show how to construct a single shallow cutting in Onlogn worst-case time, using On space. We also show that the same complexity suffices to construct simultaneously a logarithmic number of shallow cuttings on the input pointset. Our algorithms are optimal in the comparison and algebraic-comparison models, and constitute an important step forwards as the first improvement over previous deterministic polynomial-time guarantees by Matoušek (Comput Geom 2(3):169-186, 1992) and Agarwal et al. (SIAM J Comput 29(3):912-953, 2000) matching the complexity of the optimal deteministic algorithm for the more general 3-d halfspace ranges by Chan and Tsakalidis (Discrete Comput Geom 56(4):866-881, 2016). Our methods yield worst-case efficient preprocessing algorithms for a series of important orthogonal range searching problems in the pointer machine and the word-RAM models, where such shallow cuttings are utilized to support queries efficiently. [ABSTRACT FROM AUTHOR]

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). [ABSTRACT FROM AUTHOR]