Optimal deterministic shallow cuttings for 3D 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 a set of n three-dimensional points, they construct optimal size (single and multiple) shallow cuttings for orthogonal dominance ranges. In particular, we show how to construct a single shallow cutting in O(n log n) worst case time, using O(n) space. We also show how to construct in the same complexity, a logarithmic number of shallow cuttings of the input simultaneously. Our algorithms are optimal in the comparison and the algebraic comparison models, and they are an important step forward, since only polynomial guarantees were previously achieved for the deterministic construction of shallow cuttings, even in three dimensions. In fact, our methods yield the first 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 utilised to support the queries efficiently.