An Empirical Evaluation of Preconditioning Data for Accelerating Convex Hull Computations

TL;DR

This paper introduces a linear-time preconditioning algorithm for 2D convex hull computations that reduces data points and accelerates the process, achieving at least four times faster performance than previous methods.

Contribution

It presents a novel data preconditioning technique that simplifies and speeds up convex hull calculations without explicit sorting, compatible with linear-time algorithms.

Findings

Achieves at least fourfold acceleration over existing preconditioning methods.

Operates in linear time without explicit data sorting.

Produces an ordered set of points suitable for direct pipeline into O(n) convex hull algorithms.

Abstract

The convex hull describes the extent or shape of a set of data and is used ubiquitously in computational geometry. Common algorithms to construct the convex hull on a finite set of n points (x,y) range from O(nlogn) time to O(n) time. However, it is often the case that a heuristic procedure is applied to reduce the original set of n points to a set of s < n points which contains the hull and so accelerates the final hull finding procedure. We present an algorithm to precondition data before building a 2D convex hull with integer coordinates, with three distinct advantages. First, for all practical purposes, it is linear; second, no explicit sorting of data is required and third, the reduced set of s points is constructed such that it forms an ordered set that can be directly pipelined into an O(n) time convex hull algorithm. Under these criteria a fast (or O(n)) pre-conditioner in principle creates a fast convex hull (approximately O(n)) for an arbitrary set of points. The paper empirically evaluates and quantifies the acceleration generated by the method against the most common convex hull algorithms. An extra acceleration of at least four times when compared to previous existing preconditioning methods is found from experiments on a dataset.