Faster Geometric Algorithms via Dynamic Determinant Computation

TL;DR

This paper introduces dynamic determinant algorithms that significantly accelerate geometric computations like convex hulls and volume calculations by reusing previous determinant information, achieving up to 78-fold speed-ups.

Contribution

The paper presents novel dynamic determinant algorithms with quadratic and linear complexity for geometric problems, outperforming existing methods through extensive experiments.

Findings

20x speed-up in volume computation in dimension 6

78x speed-up in point location in dimension 11

Demonstrates superiority over state-of-the-art determinant algorithms

Abstract

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.