arXiv cs/04070202004-07-08cs.CG

高维最小包围多面体

Minimum Enclosing Polytope in High Dimensions

Rina Panigrahy

备注：: Submitted to SODA05

摘要：

研究了在高维$d$维空间中用给定任意形状的最小封闭多面体覆盖给定的一组$n$点的问题。我们提出的算法，适用于一个大家庭的形状，提供要么只翻译，没有旋转是允许的，或只旋转一个固定点是允许的，也就是说，只允许缩放和平移一个给定的形状，或缩放和旋转的形状周围的一个固定点。我们的算法从一个被猜测为最佳大小的多面体开始，并根据贪婪原则迭代地移动它：简单地将当前多面体直接移动到任何外部点，直到它接触表面。为了计算最小包围球，这给出了一个简单的贪心算法，其运行时间为$O(nd/\eps)$产生半径为最优值$1+\eps$倍的球。当只允许平移时，这个简单的原理推广到任意凸形状，最多需要$O(1/\eps^2)$迭代。我们的算法意味着{ \em核心集}的大小$O(1/\eps^2)$不仅存在最小封闭球，但也为任何凸形状与一个固定的方向。{ \em Core-Set}是$poly(1/\eps)$点的一个小子集，其最小包围多面体几乎与原始点的最小包围多面体一样大。虽然我们无法结合我们的技术，一般形状的平移和旋转，对于最小圆柱体的问题，我们给出了一个类似于在\cite{HV03}算法，但具有改进的运行时间的$2^{O(\frac{1}{\eps^2}\log \frac{1}{\eps})} nd$。

英文摘要：

We study the problem of covering a given set of $n$ points in a high, $d$-dimensional space by the minimum enclosing polytope of a given arbitrary shape. We present algorithms that work for a large family of shapes, provided either only translations and no rotations are allowed, or only rotation about a fixed point is allowed; that is, one is allowed to only scale and translate a given shape, or scale and rotate the shape around a fixed point. Our algorithms start with a polytope guessed to be of optimal size and iteratively moves it based on a greedy principle: simply move the current polytope directly towards any outside point till it touches the surface. For computing the minimum enclosing ball, this gives a simple greedy algorithm with running time $O(nd/\eps)$ producing a ball of radius $1+\eps$ times the optimal. This simple principle generalizes to arbitrary convex shape when only translations are allowed, requiring at most $O(1/\eps^2)$ iterations. Our algorithm implies that {\em core-sets} of size $O(1/\eps^2)$ exist not only for minimum enclosing ball but also for any convex shape with a fixed orientation. A {\em Core-Set} is a small subset of $poly(1/\eps)$ points whose minimum enclosing polytope is almost as large as that of the original points. Although we are unable to combine our techniques for translations and rotations for general shapes, for the min-cylinder problem, we give an algorithm similar to the one in \cite{HV03}, but with an improved running time of $2^{O(\frac{1}{\eps^2}\log \frac{1}{\eps})} nd$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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