A Randomized Approach to Geometric Problems

这个项目解决的中心问题是随机化与几何学的综合。我们建议研究这种综合的理论和实践意义。在几何学的背景下，传统的随机化研究--就像在随机几何学中--研究当所讨论的几何元素从良好的分布中随机选择时，各种感兴趣的量的期望值。不幸的是，这种方法只有有限的算法适用性，因为实际发现的分布并不总是表现良好。另一方面，这个项目中的方法不假设输入，只有元素被选择的顺序被认为是随机的。更准确地说，这个项目研究了当给定输入集中的几何元素被选择并以随机顺序放置在它们已经指定的位置时，配置的随机演变。实践证明，这种方法可以帮助人们设计出高效、健壮的几何问题算法。这些算法的一个显著特点是它们是随机化的，而且是递增的：它们通过考虑几何元素来进行，一次一个，但按随机顺序进行。这个项目进一步发展了基本的理论，并利用它来设计有效的算法来解决计算几何、图形学和机器人运动规划中的各种几何问题。对齐

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