Optimal-Cost Construction of Shallow Cuttings for 3-D Dominance Ranges in the I/O-Model

TL;DR

This paper presents an optimal I/O-efficient algorithm for constructing shallow cuttings in 3-D dominance range problems, improving the efficiency of spatial data structures in external memory models.

Contribution

It introduces the first optimal-cost construction algorithm for shallow cuttings in the I/O-model, reducing I/O operations for 3-D dominance range data structures.

Findings

I/O complexity of the construction algorithm is optimal at O((N/B) log_{M/B}(N/B)).

Applications include faster offline 3-D dominance reporting.

Enables more efficient data structures for 3-D range searching.

Abstract

Shallow cuttings are a fundamental tool in computational geometry and spatial databases for solving offline and online range searching problems. For a set $P$ of $N$ points in 3-D, at SODA'14, Afshani and Tsakalidis designed an optimal $O(N\log_2N)$ time algorithm that constructs shallow cuttings for 3-D dominance ranges in internal memory. Even though shallow cuttings are used in the I/O-model to design space and query efficient range searching data structures, an efficient construction of them is not known till now. In this paper, we design an optimal-cost algorithm to construct shallow cuttings for 3-D dominance ranges. The number of I/Os performed by the algorithm is $O\left(\frac{N}{B}\log_{M/B}\left(\frac{N}{B}\right) \right)$, where $B$ is the block size and $M$ is the memory size. As two applications of the optimal-cost construction algorithm, we design fast algorithms for offline 3-D dominance reporting and offline 3-D approximate dominance counting. We believe that our algorithm will find further applications in offline 3-D range searching problems and in improving construction cost of data structures for 3-D range searching problems.