Computational Geometry with Probabilistically Noisy Primitive Operations

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.

• The paper introduces error-tolerant algorithms using path-guided pushdown random walks to efficiently navigate DAG decision structures under noisy queries.
• It demonstrates that classic geometric problems like point-location, convex hulls, and Delaunay triangulations achieve O(n log n) time even with probabilistic errors.
• The study highlights practical applications in settings such as quantum computing, showing that robust data structures can significantly minimize noise overhead.

Overview of "Computational Geometry with Probabilistically Noisy Primitive Operations"

The paper "Computational Geometry with Probabilistically Noisy Primitive Operations" by David Eppstein, Michael T. Goodrich, and Vinesh Sridhar extends the investigation of error-tolerant algorithms to the field of computational geometry. The study is informed by past research on noisy sorting, and it introduces novel techniques for geometric problems under the assumption that basic operations may fail with a certain probability. This research has implications for both classical computational models as well as emerging computational paradigms such as quantum computing. The focus of the study is on computational geometry algorithms with inputs subject to probabilistic errors in primitive operations, especially Boolean geometric queries.

Key Contributions

The authors introduce a new technique termed "path-guided pushdown random walks," a generalization of noisy binary search, which enables efficient navigation of decision structures modeled by directed acyclic graphs (DAG). This method was applied successfully across different geometric problems, yielding time-efficient solutions despite the presence of noise in primitive operations.

Numerical Results and Claims

The study presents strong numerical results where computational geometry problems, such as point-location, plane-sweep, convex hulls, and Delaunay triangulations, are resolved in optimal time with high probability under the noisy scenario. Specifically, the paper demonstrates the feasibility of achieving $O(n \log n)$ time complexity for many of these problems, paralleling the time complexity observed in deterministic settings but under random noise.

Implications

• Practical Applications: The paper highlights the applicability of these techniques to practical scenarios where imprecision arises from hardware limitations or external influences, such as quantum computing platforms where queries may have an inherent probability of failing.
• Theoretical Insight: It contributes to the theoretical understanding of error-tolerant algorithms by showing that with sophisticated data structures and algorithms, the overhead induced by noise can be minimized significantly.

Future Directions

The authors suggest that the results can potentially be extended beyond the specified geometric algorithms to other computational contexts, particularly in graph algorithms. The exploration of alternative noise models and their impact on the complexity of algorithms remains an open field for future research.

Overall, this work constitutes a significant advancement by addressing computational geometry’s resilience to probabilistic errors and sets a solid foundation for future studies in error-tolerant algorithm design.