On properties of floating point arithmetics: numerical stability and the cost of accurate computations
Exact geometric computation in LEDA
Proceedings of the eleventh annual symposium on Computational geometry
Static analysis yields efficient exact integer arithmetic for computational geometry
ACM Transactions on Graphics (TOG)
Towards exact geometric computation
Computational Geometry: Theory and Applications - Special issue: computational geometry, theory and applications
Primitives for the manipulation of general subdivisions and the computation of Voronoi
ACM Transactions on Graphics (TOG)
Robust Proximity Queries: An Illustration of Degree-Driven Algorithm Design
SIAM Journal on Computing
Selected papers from the 12th annual symposium on Computational Geometry
Efficient algorithms for line and curve segment intersection using restricted predicates
Computational Geometry: Theory and Applications
Robust Plane Sweep for Intersecting Segments
SIAM Journal on Computing
Pracniques: further remarks on reducing truncation errors
Communications of the ACM
Interval arithmetic yields efficient dynamic filters for computational geometry
Discrete Applied Mathematics - Special issue 14th European workshop on computational geometry CG'98 Selected papers
Linear Time Euclidean Distance Algorithms
IEEE Transactions on Pattern Analysis and Machine Intelligence
IEEE Transactions on Pattern Analysis and Machine Intelligence
Intersecting Red and Blue Line Segments in Optimal Time and Precision
JCDCG '00 Revised Papers from the Japanese Conference on Discrete and Computational Geometry
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
Faster core-set constructions and data-stream algorithms in fixed dimensions
Computational Geometry: Theory and Applications
Parallel Banding Algorithm to compute exact distance transform with the GPU
Proceedings of the 2010 ACM SIGGRAPH symposium on Interactive 3D Graphics and Games
Can nearest neighbor searching be simple and always fast?
ESA'11 Proceedings of the 19th European conference on Algorithms
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Geometric algorithms use numerical computations to perform geometric tests, so correct algorithms may produce erroneous results if insufficient arithmetic precision is available. Liotta, Preparata, and Tamassia, in 1999, suggested that algorithm design, which traditionally considers running time and memory space, could also consider precision as a resource. They demonstrated that the Voronoi diagram of n sites on a U × U grid could be rounded to answer nearest neighbor queries on the same grid using only double precision. They still had to compute the Voronoi diagram before rounding, which requires the quadruple-precision InCircle test. We develop a "degree-2 Voronoi diagram" that can be computed using only double precision by a randomized incremental construction in O(n log n log U) expected time and O(n) expected space. Our diagram also answers nearest neighbor queries, even though it doesn't even use sufficient precision to determine a Delaunay triangulation.