On k-hulls and related problems
SIAM Journal on Computing
On-line construction of the convex hull of a simple polyline
Information Processing Letters
Dynamic planar convex hull operations in near-logarithmic amortized time
Journal of the ACM (JACM)
Design of Dynamic Data Structures
Design of Dynamic Data Structures
Optimization over k-set Polytopes and Efficient k-set Enumeration
WADS '99 Proceedings of the 6th International Workshop on Algorithms and Data Structures
On k-Nearest Neighbor Voronoi Diagrams in the Plane
IEEE Transactions on Computers
Divide and Conquer Method for k-Set Polygons
Computational Geometry and Graph Theory
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Given a set V of n points in the plane, we introduce a new number of k-sets that is an invariant of V: the number of k-sets of a convex inclusion chain of V. A convex inclusion chain of V is an ordering (v1, v2, ..., vn) of the points of V such that no point of the ordering belongs to the convex hull of its predecessors. The k-sets of such a chain are then the distinct k-sets of all the subsets {v1, ..., vi}, for all i in {k+1, ..., n}. We show that the number of these k-sets depends only on V and not on the chosen convex inclusion chain. Moreover, this number is surprisingly equal to the number of regions of the order-k Voronoi diagram of V. As an application, we give an efficient on-line algorithm to compute the k-sets of the vertices of a simple polygonal line, no vertex of which belonging to the convex hull of its predecessors on the line.