概率分布点集的形状拟合
Shape Fitting on Point Sets with Probability Distributions
摘要：
一个典型的计算几何问题开始：考虑R^d中n个点的集合P。然而，今天的许多应用程序使用的输入并不是精确已知的，例如当数据被感测并且具有某种已知的误差模型时。如果我们并不确切地知道集合P，而是我们有一个概率分布mu_p来控制P中每个点p的位置，那会怎么样呢？考虑一组（非固定）点P，设mu_P为该组点的概率分布。我们研究了几个措施（例如半径的最小封闭球，或面积的最小封闭盒）相对于mu_P。这些问题的解决方案不，在传统的情况下，由一个单一的答案，而是一个分布的答案。我们描述了几个数据结构，近似分布的形状拟合问题的答案。我们提供了简单而有效的随机算法来计算所有这些数据结构，这些算法易于实现和实用。我们提供了一些实验结果来断言这一点。我们还为这些数据结构中的一些提供了更复杂的确定性算法，这些数据结构在n和1/eps的时间多项式中运行，其中eps是近似因子。
英文摘要：
A typical computational geometry problem begins: Consider a set P of n points in R^d. However, many applications today work with input that is not precisely known, for example when the data is sensed and has some known error model. What if we do not know the set P exactly, but rather we have a probability distribution mu_p governing the location of each point p in P? Consider a set of (non-fixed) points P, and let mu_P be the probability distribution of this set. We study several measures (e.g. the radius of the smallest enclosing ball, or the area of the smallest enclosing box) with respect to mu_P. The solutions to these problems do not, as in the traditional case, consist of a single answer, but rather a distribution of answers. We describe several data structures that approximate distributions of answers for shape fitting problems. We provide simple and efficient randomized algorithms for computing all of these data structures, which are easy to implement and practical. We provide some experimental results to assert this. We also provide more involved deterministic algorithms for some of these data structures that run in time polynomial in n and 1/eps, where eps is the approximation factor.
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