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

2026-03-22

AI Summary

This work proposes the first external memory algorithm for constructing shallow-cutting structures for three-dimensional dominance ranges that achieves the theoretically optimal I/O complexity of $O\left( \frac{N}{B} \log_{M/B} \left( \frac{N}{B} \right) \right)$. By integrating block-based processing and cache-aware strategies with shallow-cutting theory from computational geometry, the algorithm efficiently constructs the data structure under limited internal memory. It overcomes the memory bottleneck inherent in handling large-scale 3D geometric data and is successfully applied to offline 3D dominance reporting and approximate counting tasks, achieving a synergistic optimization of space efficiency and query performance.

Technology Category

• Computer Vision: 3D Computer Vision
• Search and Optimization: Combinatorial Optimization
• Data Mining & Knowledge Management: Data Compression

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Bridging structured and unstructured data

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.

Problem

• shallow cuttings
• 3-D dominance ranges
• I/O-model
• range searching
• optimal-cost construction

Innovation

• shallow cuttings
• I/O-model
• 3-D dominance ranges
• optimal-cost construction
• external memory algorithms