Computational Geometry with Probabilistically Noisy Primitive Operations

TL;DR

This paper introduces a new probabilistic approach to computational geometry algorithms that tolerate noisy primitive operations, enabling robust solutions for classic problems under uncertainty.

Contribution

It proposes a novel path-guided pushdown random walk technique that generalizes noisy sorting results to computational geometry, achieving optimal-time solutions under noise.

Findings

Successfully applied to point-location, convex hulls, Delaunay triangulations

Achieves optimal time complexity with high probability

Extends noisy primitive handling to multiple geometric algorithms

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.