基于动态行列式计算的快速几何算法
Faster Geometric Algorithms via Dynamic Determinant Computation
摘要：
行列式及其符号的计算是凸包、体积、点定位等重要几何算法的核心。随着计算空间的维度增长，这些计算消耗了总计算时间的更高百分比。本文研究了几何算法中出现的行列式序列。通过使用该序列中先前计算的信息来加速单个行列式的计算。我们提出了两种动态行列式算法，当用于凸包和体积计算时具有二次算术复杂度，当用于点定位问题时具有线性算术复杂度。我们实现了所提出的算法，并进行了广泛的实验分析。一方面，我们的分析作为一个最先进的行列式算法和实现的性能研究。另一方面，我们证明了我们的方法优于最先进的实现的行列式和几何算法。我们的实验结果包括20和78倍的速度分别在体积和点定位计算在6和11维。
英文摘要：
The computation of determinants or their signs is the core procedure in many important geometric algorithms, such as convex hull, volume and point location. As the dimension of the computation space grows, a higher percentage of the total computation time is consumed by these computations. In this paper we study the sequences of determinants that appear in geometric algorithms. The computation of a single determinant is accelerated by using the information from the previous computations in that sequence. We propose two dynamic determinant algorithms with quadratic arithmetic complexity when employed in convex hull and volume computations, and with linear arithmetic complexity when used in point location problems. We implement the proposed algorithms and perform an extensive experimental analysis. On one hand, our analysis serves as a performance study of state-of-the-art determinant algorithms and implementations. On the other hand, we demonstrate the supremacy of our methods over state-of-the-art implementations of determinant and geometric algorithms. Our experimental results include a 20 and 78 times speed-up in volume and point location computations in dimension 6 and 11 respectively.
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