Faster Geometric Algorithms via Dynamic Determinant Computation

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.Comment: 29 pages, 8 figures, 3 table

Faster geometric algorithms via dynamic determinant computation

Determinant computation is the core procedure in many important geometric algorithms, such as convex hull computations and point locations. As the dimension of the computation space grows, a higher percentage of the computation time is consumed by these predicates. In this paper we study the sequences of determinants that appear in geometric algorithms. We use dynamic determinant algorithms to speed-up the computation of each predicate by using information from previously computed predicates. We propose two dynamic determinant algorithms with quadratic complexity when employed in convex hull computations, and with linear complexity when used in point location problems. Moreover, we implement them and perform an experimental analysis. Our implementations outperform the state-of-the-art determinant and convex hull implementations in most of the tested scenarios, as well as giving a speed-up of 78 times in point location problems. © 2012 Springer-Verlag