A Rank-based Convex Hull method for Dense Data Sets

TL;DR

This paper introduces a new convex hull algorithm for dense 2D integer point sets that uses ranking to avoid sorting, achieving near O(n) complexity with practical efficiency for image processing applications.

Contribution

The paper presents a novel rank-based convex hull method that reduces complexity by avoiding sorting and leverages data density and bit length for efficiency.

Findings

Achieves O(n+m) complexity for dense point sets.

Effective for image processing with densities around 3%.

Reduces time complexity further using data bit length p.

Abstract

A novel 2-D method for computing the convex hull of a sufficiently dense set of n integer points is introduced. The approach employs a ranking function that avoids sorting the points directly thus reducing the overall time complexity. The ranked points create a simple polygonal chain from which the Convex Hull can be found using a suitable O(n) method. The result is achieved by placing a bound on the density (or ratio) of points to m, where m is the maximum value of the ranking function required to represent the set of points yielding an O(n+m) method. A fast method is then developed based on the bit length, p, of the data set which reduces this time further. The required conditions are easily satisfied by image processing methods which determine the Hulls of polygonal regions where the densities are in the range of 3%. Our experiments on a range problem domains show that this is not atypical. Since the complexity of the method is related to the bit size p for current machines (p=32, p=64) the method is for all practical purposes O(n). A short proof is provided.