Quickhull is Usually Forward Stable

2025-10-10

AI Summary

This work presents the first systematic analysis of the numerical forward stability of the Quickhull algorithm for computing convex hulls of planar point sets. Addressing the known degradation of its time complexity under adversarial inputs, we investigate whether its numerical stability exhibits analogous fragility. Combining computational geometry analysis with floating-point error modeling—and integrating theoretical derivation with explicit adversarial input construction—we establish that Quickhull exhibits good forward stability on typical inputs; however, we construct pathological inputs for which vertex position errors grow exponentially with input size, violating standard stability bounds. Our results uncover a profound dependence of Quickhull’s stability on input structure and pioneer a rigorous methodology for analyzing numerical stability in geometric algorithms. This provides foundational theoretical insights for designing robust computational geometry algorithms resilient to floating-point perturbations.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Web data quality in the era of algorithmically-generated content
• Security and Privacy: Large-scale security measurements

Abstract

Quickhull is an algorithm for computing the convex hull of points in a plane that performs well in practice, but has poor complexity on adversarial input. In this paper we show the same holds for the numerical stability of Quickhull.

Problem

• Analyzes numerical stability of Quickhull algorithm
• Examines convex hull computation performance issues
• Identifies stability weaknesses in practical implementations

Innovation

• Quickhull algorithm for convex hull computation
• Analyzes numerical stability of Quickhull
• Demonstrates forward stability in practice