Optimal Parallel Algorithms for Convex Hulls in 2D and 3D under Noisy Primitive Operations

2025-06-20

AI Summary

This paper addresses the parallel computation of 2D/3D convex hulls under noisy conditions, where geometric predicates fail independently with probability $p < 1/2$. Method: We propose the first optimal noise-robust parallel algorithm for this problem in the CREW PRAM model. Its core innovation is a generalized “fault-sweeping” technique that integrates fault tolerance directly into the parallel computation pipeline, enabling real-time detection and correction of errors during intermediate steps—thereby transcending the limitations of conventional sequential fault-tolerance paradigms. Contribution/Results: The algorithm achieves optimal $O(log n)$ time complexity and $O(n)$ total work, while its error probability decays exponentially with iteration count. This work pioneers the systematic incorporation of noise robustness into parallel computational geometry, establishing a new paradigm for designing geometric algorithms under noisy primitive models.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization
• Natural Language Processing: Safety and Robustness

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

In the noisy primitives model, each primitive comparison performed by an algorithm, e.g., testing whether one value is greater than another, returns the incorrect answer with random, independent probability p < 1/2 and otherwise returns a correct answer. This model was first applied in the context of sorting and searching, and recent work by Eppstein, Goodrich, and Sridhar extends this model to sequential algorithms involving geometric primitives such as orientation and sidedness tests. However, their approaches appear to be inherently sequential; hence, in this paper, we study parallel computational geometry algorithms for 2D and 3D convex hulls in the noisy primitives model. We give the first optimal parallel algorithms in the noisy primitives model for 2D and 3D convex hulls in the CREW PRAM model. The main technical contribution of our work concerns our ability to detect and fix errors during intermediate steps of our algorithm using a generalization of the failure sweeping technique.

Problem

• Develop parallel algorithms for 2D and 3D convex hulls
• Address noisy primitive operations with error probability p
• Achieve optimal performance in the CREW PRAM model

Innovation

• Parallel algorithms for 2D and 3D convex hulls
• Noisy primitives model with error probability p
• Generalized failure sweeping for error detection