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

Computability and analysis: the legacy of Alan Turing

We discuss the legacy of Alan Turing and his impact on computability and analysis.Comment: 49 page

Can a computer be "pushed" to perform faster-than-light?

We propose to "boost" the speed of communication and computation by immersing the computing environment into a medium whose index of refraction is smaller than one, thereby trespassing the speed-of-light barrier.Comment: 7 pages, 1 figure, presented at the UC10 Hypercomputation Workshop "HyperNet 10" at The University of Tokyo on June 22, 201