Coordinate representation of order types requires exponential storage
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Reverse search for enumeration
Discrete Applied Mathematics - Special volume: first international colloquium on graphs and optimization (GOI), 1992
On the crossing number of complete graphs
Proceedings of the eighteenth annual symposium on Computational geometry
A lower bound on the number of triangulations of planar point sets
Computational Geometry: Theory and Applications
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Note: Geometric drawings of Kn with few crossings
Journal of Combinatorial Theory Series A
Abstract order type extension and new results on the rectilinear crossing number
Computational Geometry: Theory and Applications - Special issue on the 21st European workshop on computational geometry (EWCG 2005)
Decompositions, partitions, and coverings with convex polygons and pseudo-triangles
MFCS'06 Proceedings of the 31st international conference on Mathematical Foundations of Computer Science
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We extend the order type data base of all realizable order types in the plane to point sets of cardinality 11. More precisely, we provide a complete data base of all combinatorial different sets of up to 11 points in general position in the plane. In addition, we develop a novel and efficient method for a complete extension to order types of size 12 and more in an abstract sense, that is, without the need to store or realize the sets. The presented method is well suited for independent computations. Thus, time intensive investigations benefit from the possibility of distributed computing.Our approach has various applications to combinatorial problems which are based on sets of points in the plane. This includes classic problems like searching for (empty) convex k-gons ('happy end problem'), decomposing sets into convex regions, counting structures like triangulations or pseudo-triangulations, minimal crossing numbers, and more. We present some improved results to all these problems. As an outstanding result we have been able to determine the exact rectilinear crossing number of the complete graph Kn for up to n = 17, the largest previous range being n = 12, and slightly improved the asymptotic upper bound.