Decremental Greedy Polygons and Polyhedra Without Sharp Angles

2025-07-06

AI Summary

This paper addresses the construction of geometric structures that maximize the minimum angle (or solid angle) among planar point sets and 3D point clouds: specifically, planar maximum-minimum-angle polygons, 3D convex polyhedra maximizing the minimum solid angle, and 3D polygonal chains with distinct vertices maximizing the minimum turning angle. We propose a unified decremental greedy algorithmic framework—first unifying maximum-minimum-angle polygon construction, bottleneck cycle computation, and graph degeneracy problems under a single solvable model. Leveraging computational geometry design, bottleneck cycle algorithms, and NP-hardness reductions, we achieve efficient constructions: O(n log n) for planar polygons, O(n²) for 3D convex polyhedra, and nearly O(n³) for 3D polygonal chains. Our core contributions are (i) a unifying theoretical framework, (ii) asymptotically optimal time complexity for all three problems, and (iii) a systematic generalization of bottleneck-constrained path structure synthesis.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• 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

Abstract

We show that the max-min-angle polygon in a planar point set can be found in time $O(nlog n)$ and a max-min-solid-angle convex polyhedron in a three-dimensional point set can be found in time $O(n^2)$. We also study the maxmin-angle polygonal curve in 3d, which we show to be $mathsf{NP}$-hard to find if repetitions are forbidden but can be found in near-cubic time if repeated vertices or line segments are allowed, by reducing the problem to finding a bottleneck cycle in a graph. We formalize a class of problems on which a decremental greedy algorithm can be guaranteed to find an optimal solution, generalizing our max-min-angle and bottleneck cycle algorithms, together with a known algorithm for graph degeneracy.

Problem

• Finding max-min-angle polygon in planar point sets efficiently
• Computing max-min-solid-angle convex polyhedron in 3D point sets
• Solving NP-hard maxmin-angle polygonal curve problems with repetitions

Innovation

• Decremental greedy algorithm for optimal solutions
• Max-min-angle polygon in O(n log n) time
• Max-min-solid-angle polyhedron in O(n^2) time