Planar point location using persistent search trees
Communications of the ACM
Optimal point location in a monotone subdivision
SIAM Journal on Computing
A fast planar partition algorithm, I
Journal of Symbolic Computation
Computational Geometry: Theory and Applications
On the exact worst case query complexity of planar point location
Journal of Algorithms
The design and implementation of panar maps in CGAL
Journal of Experimental Algorithmics (JEA)
Incremental constructions con BRIO
Proceedings of the nineteenth annual symposium on Computational geometry
Location of a point in a planar subdivision and its applications
STOC '76 Proceedings of the eighth annual ACM symposium on Theory of computing
On translating a set of rectangles
STOC '80 Proceedings of the twelfth annual ACM symposium on Theory of computing
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
An experimental study of point location in planar arrangements in CGAL
Journal of Experimental Algorithmics (JEA)
Hi-index | 0.00 |
We present a major revamp of the point-location data structure for general two-dimensional subdivisions via randomized incremental construction, implemented in Cgal, the Computational Geometry Algorithms Library. We can now guarantee that the constructed directed acyclic graph $\mathcal G$ is of linear size and provides logarithmic query time. Via the construction of the Voronoi diagram for a given point set S of size n, this also enables nearest-neighbor queries in guaranteed O(logn) time. Another major innovation is the support of general unbounded subdivisions as well as subdivisions of two-dimensional parametric surfaces such as spheres, tori, cylinders. The implementation is exact, complete, and general, i.e., it can also handle non-linear subdivisions. Like the previous version, the data structure supports modifications of the subdivision, such as insertions and deletions of edges, after the initial preprocessing. A major challenge is to retain the expected O(n logn) preprocessing time while providing the above (deterministic) space and query-time guarantees. We describe efficient preprocessing algorithms, which explicitly verify the length $\mathcal L$ of the longest query path. However, instead of using $\mathcal L$, our implementation is based on the depth $\mathcal D$ of $\mathcal G$. Although we prove that the worst case ratio of $\mathcal D$ and $\mathcal L$ is Θ(n/logn), we conjecture, based on our experimental results, that this solution achieves expected O(n logn) preprocessing time.