Optimal Deterministic Shallow Cuttings for 3-d Dominance Ranges

Abstract

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 $O\left( n\log n\right)$ worst-case time, using $O\left( n\right)$ 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.