On the predictability of the number of convex vertices

TL;DR

This paper discovers that the number of convex vertices in natural datasets can be predicted within a specific range based on dataset size, aiding algorithm optimization and data modeling.

Contribution

It introduces an empirical range prediction for convex vertices in natural datasets, highlighting the importance of bounded data models for algorithm testing.

Findings

Number of convex vertices scales as approximately 2.35 N^0.091 to 19.19 N^0.091

Range applies to natural datasets, not worst-case scenarios

Implications for data pre-allocation and algorithm evaluation

Abstract

Convex hulls are a fundamental geometric tool used in a number of algorithms. As a side-effect of exhaustive tests for an algorithm for which a convex hull computation was the first step, interesting experimental results were found and are the sunject of this paper. They establish that the number of convex vertices of natural datasets can be predicted, if not precisely at least within a defined range. Namely it was found that the number of convex vertices of a dataset of N points lies in the range 2.35 N^0.091 <= h <= 19.19 N^0.091. This range obviously does not describe neither natural nor artificial worst-cases but corresponds to the distributions of natural data. This can be used for instance to define a starting size for pre-allocated arrays or to evaluate output-sensitive algorithms. A further consequence of these results is that the random models of data used to test convex hull algorithms should be bounded by rectangles and not as they usually are by circles if they want to represent accurately natural datasets