Geometric Systems of Unbiased Representatives
| Autor: | Banik, Aritra, Bhattacharya, Bhaswar B., Bhore, Sujoy, Martínez-Sandoval, Leonardo |
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| Rok vydání: | 2020 |
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| Druh dokumentu: | Working Paper |
| Popis: | Let $P$ be a set of points in $\mathbb{R}^d$, $B$ a bicoloring of $P$ and $\Oo$ a family of geometric objects (that is, intervals, boxes, balls, etc). An object from $\Oo$ is called balanced with respect to $B$ if it contains the same number of points from each color of $B$. For a collection $\B$ of bicolorings of $P$, a geometric system of unbiased representatives (G-SUR) is a subset $\Oo'\subseteq\Oo$ such that for any bicoloring $B$ of $\B$ there is an object in $\Oo'$ 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. Comment: Appears in the Proceedings of the 31st Canadian Conference on Computational Geometry (CCCG 2019) |
| Databáze: | arXiv |
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