Geometric Systems of Unbiased Representatives
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Let P be a set of points in ℝ^d, B a bicoloring of P and a family of geometric objects (that is, intervals, boxes, balls, etc). An object from is called balanced with respect to B if it contains the same number of points from each color of B. For a collection of bicolorings of P, a geometric system of unbiased representatives (G-SUR) is a subset '⊆ such that for any bicoloring B of there is an object in ' that is balanced with respect to B. We study the problem of finding G-SURs. We obtain general bounds on the size of G-SURs consisting of intervals, size-restricted intervals, axis-parallel boxes and Euclidean balls. We show that the G-SUR problem is NP-hard even in the simple case of points on a line and interval ranges. Furthermore, we study a related problem on determining the size of the largest and smallest balanced intervals for points on the real line with a random distribution and coloring. Our results are a natural extension to a geometric context of the work initiated by Balachandran et al. on arbitrary systems of unbiased representatives.
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