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An $O(N)$ Algorithm for Solving the Smallest Enclosing Sphere Problem in the Presence of Degeneracies

Netzer Moriya

TL;DR

The paper tackles the 3D Smallest Enclosing Sphere (SES) problem under degeneracies, where classical Welzl-like methods can fail or slow down. It introduces a hybrid projection-based framework that constructs a reduced point set $P_s$ from projections onto $K$ planes and proves key theoretical results: (i) solving SES for $P$ is equivalent to solving SES for the convex hull $P_{ch}$, (ii) degeneracies from internal points are avoided by working with $P_{ch}$, and (iii) as $K\to\infty$, $P_s$ converges to $P_{ch}$, making the method robust to degeneracies. The algorithm achieves $O(KN)$ time, which collapses to $O(N)$ when $K\ll N$, and does so without requiring expensive preprocessing or exact arithmetic while preserving geometric fidelity. These contributions provide a practically efficient and robust SES computation suitable for large datasets with degeneracies, by combining a convex-hull-centric reduction with a directional projection scheme. The work offers theoretical guarantees for equivalence and convergence and delivers practical bounds for projection counts, contributing a scalable approach to robust SES in robotics, graphics, and data analysis.

Abstract

Efficient algorithms for solving the Smallest Enclosing Sphere (SES) problem, such as Welzl's algorithm, often fail to handle degenerate subsets of points in 3D space. Degeneracies and ill-posed configurations present significant challenges, leading to failures in convergence, inaccuracies or increased computational cost in such cases. Existing improvements to these algorithms, while addressing some of these issues, are either computationally expensive or only partially effective. In this paper, we propose a hybrid algorithm designed to mitigate degeneracy while maintaining an overall computational complexity of $O(N)$. By combining robust preprocessing steps with efficient core computations, our approach avoids the pitfalls of degeneracy without sacrificing scalability. The proposed method is validated through theoretical analysis and experimental results, demonstrating its efficacy in addressing degenerate configurations and achieving high efficiency in practice.

An $O(N)$ Algorithm for Solving the Smallest Enclosing Sphere Problem in the Presence of Degeneracies

TL;DR

The paper tackles the 3D Smallest Enclosing Sphere (SES) problem under degeneracies, where classical Welzl-like methods can fail or slow down. It introduces a hybrid projection-based framework that constructs a reduced point set from projections onto planes and proves key theoretical results: (i) solving SES for is equivalent to solving SES for the convex hull , (ii) degeneracies from internal points are avoided by working with , and (iii) as , converges to , making the method robust to degeneracies. The algorithm achieves time, which collapses to when , and does so without requiring expensive preprocessing or exact arithmetic while preserving geometric fidelity. These contributions provide a practically efficient and robust SES computation suitable for large datasets with degeneracies, by combining a convex-hull-centric reduction with a directional projection scheme. The work offers theoretical guarantees for equivalence and convergence and delivers practical bounds for projection counts, contributing a scalable approach to robust SES in robotics, graphics, and data analysis.

Abstract

Efficient algorithms for solving the Smallest Enclosing Sphere (SES) problem, such as Welzl's algorithm, often fail to handle degenerate subsets of points in 3D space. Degeneracies and ill-posed configurations present significant challenges, leading to failures in convergence, inaccuracies or increased computational cost in such cases. Existing improvements to these algorithms, while addressing some of these issues, are either computationally expensive or only partially effective. In this paper, we propose a hybrid algorithm designed to mitigate degeneracy while maintaining an overall computational complexity of . By combining robust preprocessing steps with efficient core computations, our approach avoids the pitfalls of degeneracy without sacrificing scalability. The proposed method is validated through theoretical analysis and experimental results, demonstrating its efficacy in addressing degenerate configurations and achieving high efficiency in practice.

Paper Structure

This paper contains 40 sections, 8 theorems, 12 equations, 1 table.

Table of Contents

  1. Introduction
  2. Background and Related Work
  3. Welzl's Algorithm Overview
  4. Degenerate Configurations and Challenges
  5. Welzl's Algorithm: Modifications and Robustness Improvements
  6. 1. Lower-Dimensional Handling
  7. 2. Symbolic Perturbation Techniques
  8. 3. Convex Hull Preprocessing
  9. 4. Exact Arithmetic and Robust Geometric Predicates
  10. Analysis of Computational Cost
  11. Proposed Method
  12. Preliminary Definitions and Lemmas
  13. Proof of the Equivalence of SES for a Set of Points $P$ and Its Convex Hull $P_{ch}$
  14. Theorem
  15. Proof
  16. ...and 25 more sections

Key Result

Lemma 1

Let $P \subset \mathbb{R}^3$, and let $P_{ch}$ be its convex hull. For any linear projection $\pi: \mathbb{R}^3 \to \mathbb{R}^2$, the projection $\pi(P_{ch})$ is the convex hull of $\pi(P)$.

Theorems & Definitions (18)

  • Definition 1: Convex Hull
  • Definition 2: Extreme Points
  • Lemma 1: Projection and Convexity
  • proof
  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 8 more