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
SCG '99 Proceedings of the fifteenth annual symposium on Computational geometry
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
Abstract order type extension and new results on the rectilinear crossing number
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
GD'04 Proceedings of the 12th international conference on Graph Drawing
Improved upper bounds on the reflexivity of point sets
Computational Geometry: Theory and Applications
On the Heilbronn optimal configuration of seven points in the square
ADG'08 Proceedings of the 7th international conference on Automated deduction in geometry
Reprint of: Extreme point and halving edge search in abstract order types
Computational Geometry: Theory and Applications
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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 several of 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.