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

TL;DR

This paper introduces an $O(N)$ algorithm for the Smallest Enclosing Sphere problem that effectively handles degeneracies in 3D point sets, improving robustness and efficiency over existing methods.

Contribution

A hybrid algorithm combining preprocessing and core computations to address degeneracies in the SES problem with linear complexity.

Findings

Successfully handles degenerate point configurations

Maintains $O(N)$ computational complexity

Demonstrates high efficiency in experiments

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.