Computing Largest Subsets of Points Whose Convex Hulls have Bounded Area and Diameter

TL;DR

This paper introduces an algorithm to find the largest subset of points enclosed in a convex region with bounded area and diameter, with applications demonstrated in cancer detection.

Contribution

The paper presents a novel algorithm for computing maximum point subsets within convex regions constrained by area and diameter, with improved complexity analysis.

Findings

The new algorithm runs in $O(n^6k)$ time, slower than the $O(n^3k)$ of the diameter-only algorithm.

Experimental results compare the new method with existing algorithms on synthetic and real data.

Application in cancer detection demonstrates practical relevance of the approach.

Abstract

We study the problem of computing a convex region with bounded area and diameter that contains the maximum number of points from a given point set $P$. We show that this problem can be solved in $O(n^6k)$ time and $O(n^3k)$ space, where $n$ is the size of $P$ and $k$ is the maximum number of points in the found region. We experimentally compare this new algorithm with an existing algorithm that does the same but without the diameter constraint, which runs in $O(n^3k)$ time. For the new algorithm, we use different diameters. We use both synthetic data and data from an application in cancer detection, which motivated our research.