Using Optimization to Find Maximum Inscribed Balls and Minimum Enclosing Balls
Zeyuan Allen Zhu, Zhenyu Liao, Lorenzo Orecchia · arXiv (Cornell University) · 2014
We study two fundamental problems in computational geometry: finding the maximum inscribed ball (MaxIB) inside a polytope defined by $m$ hyperplanes in a $d$-dimensional space, and finding the minimum enclosing ball (MinEB) of a set of $n$ points in a $d$-dimensional space. We translate both geometry problems into purely algebraic optimization questions, and apply ideas from first-order convex optimizations to obtain simple and nearly-linear-time algorithms. For MaxIB, our algorithm produces a $(1-\varepsilon)$-approximate solution in time $\tilde{O}(m d \alpha / \varepsilon)$, compared to the best known running time $\tilde{O}(m d \alpha^3 / \varepsilon^3)$ of Xie, Snoeyink, and Xu [XSX06], where $\alpha \geq 1$ is the aspect ratio of the polytope. For MinEB, our algorithm produces a $(1+\varepsilon)$-approximate solution in time $\tilde{O}(n d / \sqrt{\varepsilon})$. This result matches the previous best running time, while yielding a simpler algorithm. For both problems, we provide an empirical evaluation showing the improved performance of our algorithms.