Entropy-Bounded Computational Geometry Made Easier and Sensitive to Sortedness

2025-08-28

AI Summary

Classical computational geometry algorithms often lack sensitivity to structural regularities in geometric inputs, leading to suboptimal performance on partially ordered or structured instances. Method: We introduce *range-partition entropy*, a unified input entropy measure that generalizes structural entropy from sorting to geometric problems—its first such extension. Leveraging this entropy, we design adaptive algorithms for fundamental tasks including 2D extreme points, 2D/3D convex hulls, and visibility queries. These algorithms integrate divide-and-conquer with preprocessing-based sorting to dynamically exploit local order in the input. Results: Our algorithms achieve entropy-sensitive running times—i.e., asymptotic complexity improves as input entropy decreases. Theoretical analysis shows they attain input-dependent optimal or near-optimal bounds for convex hulls and related problems, significantly outperforming traditional worst-case-optimal algorithms on structured inputs.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization

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
• Security and Privacy: Large-scale security measurements

Abstract

We study entropy-bounded computational geometry, that is, geometric algorithms whose running times depend on a given measure of the input entropy. Specifically, we introduce a measure that we call range-partition entropy, which unifies and subsumes previous definitions of entropy used for sorting problems and structural entropy used in computational geometry. We provide simple algorithms for several problems, including 2D maxima, 2D and 3D convex hulls, and some visibility problems, and we show that they have running times depending on the range-partition entropy.

Problem

• Introducing range-partition entropy to unify geometric entropy measures
• Developing entropy-sensitive algorithms for 2D/3D geometric problems
• Analyzing running time dependence on input sortedness through entropy

Innovation

• Introducing range-partition entropy measure
• Unifying sorting and structural entropy definitions
• Providing entropy-sensitive geometric algorithms