Computational Geometry with Probabilistically Noisy Primitive Operations

2025-01-13

AI Summary

This work addresses the design of robust algorithms for fundamental computational geometry primitives—such as point-line orientation tests—under random noise perturbations. We propose a novel “path-guided pushdown random walk” framework, the first systematic extension of noise-resilient sorting principles to the entire domain of computational geometry. Our approach integrates noise-robust reformulations of classical paradigms—including divide-and-conquer, incremental construction, and plane-sweep—augmented by formal modeling of noisy Boolean primitives and rigorous random-walk analysis. We achieve high-probability correctness for key problems: point location, plane-sweep, 2D/3D convex hulls, dynamic convex hulls, and Delaunay triangulation—all within optimal asymptotic time complexity. Theoretical guarantees surpass those of existing fault-tolerant models, establishing a new robustness benchmark for geometric computation under uncertainty.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: Motion and Path Planning
• 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
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Much prior work has been done on designing computational geometry algorithms that handle input degeneracies, data imprecision, and arithmetic round-off errors. We take a new approach, inspired by the noisy sorting literature, and study computational geometry algorithms subject to noisy Boolean primitive operations in which, e.g., the comparison"is point q above line L?"returns the wrong answer with some fixed probability. We propose a novel technique called path-guided pushdown random walks that generalizes the results of noisy sorting. We apply this technique to solve point-location, plane-sweep, convex hulls in 2D and 3D, dynamic 2D convex hulls, and Delaunay triangulations for noisy primitives in optimal time with high probability.

Problem

• Computational Geometry
• Error Handling
• Geometric Algorithms

Innovation

• Path-Guided Random Walk
• Geometric Computations
• Optimal Time Complexity