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.

翻译：高效解决最小包围球问题（如Welzl算法）的算法通常无法处理三维空间中的退化点集。退化与病态配置带来了重大挑战，导致此类情况下算法收敛失败、精度不足或计算成本增加。现有改进算法虽能部分解决这些问题，但要么计算代价高昂，要么效果有限。本文提出一种混合算法，旨在缓解退化问题，同时保持$O(N)$的整体计算复杂度。通过将鲁棒的预处理步骤与高效的核心计算相结合，我们的方法在保持可扩展性的同时避免了退化陷阱。通过理论分析和实验结果验证了所提方法的有效性，证明其能够处理退化配置并在实践中实现高效率。