## Abstract

In 2015, Guth proved that if S is a collection of n g-dimensional semi-algebraic sets in ℝ^{d} and if D ≥ 1 is an integer, then there is a d-variate polynomial P of degree at most D so that each connected component of ℝ^{d} \ Z(P) intersects O(n/D^{d−g}) sets from S. Such a polynomial is called a generalized partitioning polynomial. We present a randomized algorithm that computes such polynomials efficiently – the expected running time of our algorithm is linear in |S|. Our approach exploits the technique of quantifier elimination combined with that of ε-samples. We present four applications of our result. The first is a data structure for answering point-enclosure queries among a family of semi-algebraic sets in R^{d} in O(log n) time, with storage complexity and expected preprocessing time of O(n^{d+ε}). The second is a data structure for answering range search queries with semi-algebraic ranges in O(log n) time, with O(n^{t+ε}) storage and expected preprocessing time, where t > 0 is an integer that depends on d and the description complexity of the ranges. The third is a data structure for answering vertical ray-shooting queries among semi-algebraic sets in ℝ^{d} in O(log^{2} n) time, with O(n^{d+ε}) storage and expected preprocessing time. The fourth is an efficient algorithm for cutting algebraic planar curves into pseudo-segments.

Original language | English |
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Title of host publication | 35th International Symposium on Computational Geometry, SoCG 2019 |

Editors | Gill Barequet, Yusu Wang |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959771047 |

DOIs | |

State | Published - 1 Jun 2019 |

Event | 35th International Symposium on Computational Geometry, SoCG 2019 - Portland, United States Duration: 18 Jun 2019 → 21 Jun 2019 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 129 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 35th International Symposium on Computational Geometry, SoCG 2019 |
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Country/Territory | United States |

City | Portland |

Period | 18/06/19 → 21/06/19 |

### Bibliographical note

Publisher Copyright:© Pankaj K. Agarwal, Boris Aronov, Esther Ezra, and Joshua Zahl.

### Funding

Funding Pankaj K. Agarwal: P. Agarwal was supported by NSF under grants CCF-15-13816, CCF-15-46392, and IIS-14-08846, by an ARO grant W911NF-15-1-0408, and by BSF Grant 2012/229 from the U.S.-Israel Binational Science Foundation. Boris Aronov: B. Aronov was supported by NSF grants CCF-12-18791 and CCF-15-40656, and by grant 2014/170 from the US-Israel Binational Science Foundation. Esther Ezra: E. Ezra was supported by NSF CAREER under grant CCF:AF 1553354 and by Grant 824/17 from the Israel Science Foundation. Joshua Zahl: J. Zahl was supported by an NSERC Discovery grant. Pankaj K. Agarwal: P. Agarwal was supported by NSF under grants CCF-15-13816, CCF-15-46392, and IIS-14-08846, by an ARO grant W911NF-15-1-0408, and by BSF Grant 2012/229 from the U.S.-Israel Binational Science Foundation. Boris Aronov: B. Aronov was supported by NSF grants CCF-12-18791 and CCF-15-40656, and by grant 2014/170 from the US-Israel Binational Science Foundation. Esther Ezra: E. Ezra was supported by NSF CAREER under grant CCF:AF 1553354 and by Grant 824/17 from the Israel Science Foundation. Joshua Zahl: J. Zahl was supported by an NSERC Discovery grant.

Funders | Funder number |
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U.S.-Israel Binational Science Foundation | 2014/170 |

US-Israel Binational Science Foundation | AF 1553354 |

National Science Foundation | CCF-15-46392, CCF-15-13816, CCF:AF 1553354, CCF-12-18791, IIS-14-08846, CCF-15-40656 |

Army Research Office | W911NF-15-1-0408 |

Bloom's Syndrome Foundation | 2012/229 |

Natural Sciences and Engineering Research Council of Canada | |

United States-Israel Binational Science Foundation | |

Israel Science Foundation | 824/17 |

## Keywords

- Polynomial partitioning
- Quantifier elimination
- Semi-algebraic range spaces
- ε-samples