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

TL;DR

This paper develops the first optimal parallel algorithms for computing convex hulls in 2D and 3D under noisy comparison models, effectively detecting and correcting errors during computation.

Contribution

It introduces the first optimal parallel algorithms for convex hulls in noisy primitive models, extending previous sequential approaches to parallel settings.

Findings

First optimal parallel algorithms for 2D and 3D convex hulls in noisy models.

Effective error detection and correction during intermediate steps.

Extension of failure sweeping technique to parallel geometric algorithms.

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.