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A simple and efficient preprocessing step for convex hull problem

Mohammad Heydari, Ashkan Khalifeh

TL;DR

For such a set of points, it is shown that eliminating all but all but $O(\log n)$ of points can derive the same convex hull as the input set.

Abstract

The present paper is concerned with a recursive algorithm as a preprocessing step to find the convex hull of $n$ random points uniformly distributed in the plane. For such a set of points, it is shown that eliminating all but $O(\log n)$ of points can derive the same convex hull as the input set. Finally it will be shown that the running time of the algorithm is $O(n)

A simple and efficient preprocessing step for convex hull problem

TL;DR

For such a set of points, it is shown that eliminating all but all but of points can derive the same convex hull as the input set.

Abstract

The present paper is concerned with a recursive algorithm as a preprocessing step to find the convex hull of random points uniformly distributed in the plane. For such a set of points, it is shown that eliminating all but of points can derive the same convex hull as the input set. Finally it will be shown that the running time of the algorithm is $O(n)
Paper Structure (5 sections, 5 theorems, 10 equations, 3 figures, 1 algorithm)

This paper contains 5 sections, 5 theorems, 10 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Algorithm
  4. Correctness and Analysis
  5. Conclusion

Key Result

Lemma 1

Consider a set of uniformly distributed points and its bounding box in the plane, as shown in Figure fig:rec1. Connecting the point on an edge of the bounding box to a point on its adjacent edges, will form a quadrilateral inside the bounding box. The expected number of points inside the quadrilater

Figures (3)

  • Figure 1: Illustration of quadrilateral made by connecting extreme points.
  • Figure 2: Right triangles.
  • Figure 3: Illustration of execution of the algorithm.

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof