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We consider point sets in (Z^2,n) where no three points are on a
line – also called caps or arcs. For the determination of caps with maximum
cardinality and complete caps with minimum cardinality we provide integer
linear programming formulations and identify some values for small n.
We describe several algorithms for the generation of integer Heronian triangles with diameter at most n. Two of them have running time O(n^(2+ε)). We enumerate all integer Heronian triangles for n ≤ 600000
and apply the complete list on some related problems.
ACM Computing Classification System (1998): G.2, G.4.
A set of n lattice points in the plane, no three on a line and no
four on a circle, such that all pairwise distances and coordinates are integers
is called an n-cluster (in R^2). We determine the smallest 7-cluster with
respect to its diameter. Additionally we provide a toolbox of algorithms
which allowed us to computationally locate over 1000 different 7-clusters,
some of them having huge integer edge lengths. Along the way, we have
exhaustively...
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