TY - GEN

T1 - Improved implementation of point location in general two-dimensional subdivisions

AU - Hemmer, Michael

AU - Kleinbort, Michal

AU - Halperin, Dan

N1 - Funding Information:
This work has been supported in part by the 7th Framework Programme for Research of the European Commission, under FET-Open grant number 255827 (CGL—Computational Geometry Learning), by the Israel Science Foundation (grant no. 1102/11), and by the Hermann Minkowski–Minerva Center for Geometry at Tel Aviv University.

PY - 2012

Y1 - 2012

N2 - 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 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 L of the longest query path. However, instead of using L, our implementation is based on the depth D of G. Although we prove that the worst case ratio of D and L is Θ(n/logn), we conjecture, based on our experimental results, that this solution achieves expected O(n logn) preprocessing time.

AB - 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 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 L of the longest query path. However, instead of using L, our implementation is based on the depth D of G. Although we prove that the worst case ratio of D and L is Θ(n/logn), we conjecture, based on our experimental results, that this solution achieves expected O(n logn) preprocessing time.

UR - http://www.scopus.com/inward/record.url?scp=84866716540&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-33090-2_53

DO - 10.1007/978-3-642-33090-2_53

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AN - SCOPUS:84866716540

SN - 9783642330896

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 611

EP - 623

BT - Algorithms, ESA 2012 - 20th Annual European Symposium, Proceedings

T2 - 20th Annual European Symposium on Algorithms, ESA 2012

Y2 - 10 September 2012 through 12 September 2012

ER -