The algorithm design manual
Computing Two-Dimensional Integer Hulls
SIAM Journal on Computing
The Art of Computer Programming Volumes 1-3 Boxed Set
The Art of Computer Programming Volumes 1-3 Boxed Set
Algebra: An elementary text-book.--
Algebra: An elementary text-book.--
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
Reducing the coefficients of a two-dimensional integer linear constraint
IWCIA'08 Proceedings of the 12th international conference on Combinatorial image analysis
Walking in the farey fan to compute the characteristics of a discrete straight line subsegment
DGCI'13 Proceedings of the 17th IAPR international conference on Discrete Geometry for Computer Imagery
Hi-index | 0.04 |
We consider the problem of determining the rational number which best approximates the real number a and such that its denominator belongs to an interval [b,b^']. There is a related geometric problem consisting in finding the integer point lying in the vertical domain D of the form {(x,y)@?R^2|b@?x@?b^'} such that the straight line passing through the origin and through this point best approximates the straight line L of slope a passing through the origin. The computation of this point is interlinked with the computation of both the convex hulls of the integer points located above and below the straight line L respectively and lying in the vertical domain D. In the literature, many general convex hull algorithms exist, as the gift wrapping algorithm for example. However, we focus on two interesting approaches to compute these convex hulls which are especially appropriated in our special configuration. The first one mainly uses number theory and runs in O(log(b^')) time. The other is in line with computational geometry as the method proposed in 1999 by Balza-Gomez et al. [H. Balza-Gomez, J.-M. Moreau, D. Michelucci, Convex hull of grid points below a line or a convex curve, in: DGCI '99: Proceedings of the 8th International Conference on Discrete Geometry for Computer Imagery, Springer, Marne-la-Vallee, France, 1999, pp. 361-374] which runs in O(log(b^'-b)) time. We propose a new method for the computation of these convex hulls which combines number theory and computational geometry. Our method preserves the optimal time complexity and is the first being output sensitive. Indeed, we compute the convex hulls in time linear in their vertex number. Moreover, the resulting algorithm is very simple and so is suitable for implementation.