## Abstract

Let P be a set of points in Rd, B a bicoloring of P and O a family of geometric objects (that is, intervals, boxes, balls, etc). An object from O 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 (GSUR) is a subset such that for any bicoloring B of β there is an object in O0 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.

Original language | English |
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Pages | 38-43 |

Number of pages | 6 |

State | Published - 1 Jan 2019 |

Externally published | Yes |

Event | 31st Canadian Conference on Computational Geometry, CCCG 2019 - Edmonton, Canada Duration: 8 Aug 2019 → 10 Aug 2019 |

### Conference

Conference | 31st Canadian Conference on Computational Geometry, CCCG 2019 |
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Country/Territory | Canada |

City | Edmonton |

Period | 8/08/19 → 10/08/19 |

## ASJC Scopus subject areas

- Geometry and Topology
- Computational Mathematics