Algorithms in combinatorial geometry
Algorithms in combinatorial geometry
Opposite-Quadrant Depth in the Plane
Graphs and Combinatorics
Note: Points with large α-depth
Journal of Combinatorial Theory Series A
On a problem about quadrant-depth
Computational Geometry: Theory and Applications
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Given a set P of points in the plane we are interested in points that are 'deep' in the set in the sense that they have two opposite quadrants both containing many points of P. We deal with the extremal version of this problem. A pair (a, b) of numbers is admissible if every point set P contains a point p ∈ P that determines a pair (Q,Qop) of opposite quadrants, such that Q contains at least an a-fraction and Qop contains at least a b-fraction of the points of P. We provide a complete description of the set F of all admissible pairs (a, b). This amounts to identifying three line segments and a point on the boundary of F. In higher dimensions we study the maximal a, such that (a, a) is opposite orthant admissible. We show that 1/(2γ) d a d 1/γ for γ = 22d--1 2d--1. Finally we deal with a variant of the problem where the opposite pairs of orthants need not be determined by a point in P. Again we are interested in values a, such that all subsets P in ℜd admit a pair (O,Oop) of opposite orthants both containing at least an a-fraction of the points. The maximal such value is a = 1/2d. Generalizations of the problem are also discussed.