A Randomized Approach to Geometric Problems

该项目所解决的中心问题是综合 随机化与几何 我们建议研究的理论， 以及这种综合的实际意义。 一个传统 几何学背景下的随机化研究 几何学-研究各种数量的期望值 兴趣，当所讨论的几何元素是随机选择的 从一个不错的分布。 不幸的是，这种方法只有一个 有限的算法适用性，因为分布实际上 在实践中，并不总是表现良好。 另一方面 这个项目中的方法对输入没有任何假设， 选择元素的顺序应该是随机的。 更确切地说，这个项目研究了 配置，因为给定输入集中的几何元素是 选择，并把它们放在已经指定的位置，在一个随机的 秩序 事实证明，这种方法有助于设计算法 对于几何问题，这是非常有效的，和强大的。 一 这些算法的一个显著特点是， 随机化和增量：他们继续考虑几何 元素，一次一个，但顺序随机。 该项目进一步 发展基本理论，并利用它来设计高效的 计算几何中各种几何问题的算法， 图形和机器人运动规划。 理由

The central problem addressed by this project is the synthesis of randomization with geometry. We propose to study the theoretical as well as the practical implications of such a synthesis. A tradition study of randomization in the context of geometry-as in stochastic geometry - studies the expected value of various quantities of interest, when the geometric elements in question are chosen randomly from a nice distribution. Unfortunately, this approach has only a limited algorithmic applicability, because the distributions actually found in practice are not always well behaved. On the other hand, the approach in this project assumes nothing about the input, and only the order, in which the elements are chosen, is supposed to be random. More precisely, this project studies the random evolution of the configuration as the geometric elements in the given input set are chosen, and put at their already specified locations, in a random order. It turns out that, this approach helps one design algorithms for geometric problems, which are very efficient, and robust. A distinguishing characteristic of these algorithms is that they are randomized, and incremental: they proceed by considering the geometric elements, one at a time, but in a random order. This project further develop the underlying theory, and uses it to design efficient algorithms for various geometric problems in computational geometry, graphics, and robot motion planning. JUSTIFICATION