TY - JOUR
T1 - Empty pseudo-triangles in point sets
AU - Ahn, H.K.
AU - Bae, S.W.
AU - Kreveld, van, M.J.
AU - Reinbacher, I.
AU - Speckmann, B.
PY - 2011
Y1 - 2011
N2 - We study empty pseudo-triangles in a set P of n points in the plane, where an empty pseudo-triangle has its vertices at the points of P, and no points of P lie inside. We give bounds on the minimum and maximum number of empty pseudo-triangles. If P lies inside a triangle whose corners must be the convex vertices of the pseudo-triangle, then there can be between T(n2) and T(n3) empty pseudo-triangles. If the convex vertices of the pseudo-triangle are also chosen from P, this number lies between T(n3) and T(n6). If we count only star-shaped pseudo-triangles, the bounds are T(n2) and T(n5). We also study optimization problems: minimizing or maximizing the perimeter or the area over all empty pseudo-triangles defined by P. If P lies inside a triangle whose corners must be used, we can solve these problems in O(n3) time. In the general case, the running times are O(n6) for the maximization problems and O(nlogn) for the minimization problems.
AB - We study empty pseudo-triangles in a set P of n points in the plane, where an empty pseudo-triangle has its vertices at the points of P, and no points of P lie inside. We give bounds on the minimum and maximum number of empty pseudo-triangles. If P lies inside a triangle whose corners must be the convex vertices of the pseudo-triangle, then there can be between T(n2) and T(n3) empty pseudo-triangles. If the convex vertices of the pseudo-triangle are also chosen from P, this number lies between T(n3) and T(n6). If we count only star-shaped pseudo-triangles, the bounds are T(n2) and T(n5). We also study optimization problems: minimizing or maximizing the perimeter or the area over all empty pseudo-triangles defined by P. If P lies inside a triangle whose corners must be used, we can solve these problems in O(n3) time. In the general case, the running times are O(n6) for the maximization problems and O(nlogn) for the minimization problems.
U2 - 10.1016/j.dam.2011.07.026
DO - 10.1016/j.dam.2011.07.026
M3 - Article
SN - 0166-218X
VL - 159
SP - 2205
EP - 2213
JO - Discrete Applied Mathematics
JF - Discrete Applied Mathematics
IS - 18
ER -