### 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 (US) |
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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 - Jun 1 2019 |

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

City | Portland |

Period | 6/18/19 → 6/21/19 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Software

### Cite this

*35th International Symposium on Computational Geometry, SoCG 2019*[5] (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 129). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.SoCG.2019.5

**An efficient algorithm for generalized polynomial partitioning and its applications.** / Agarwal, Pankaj K.; Aronov, Boris; Ezra, Esther; Zahl, Joshua.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*35th International Symposium on Computational Geometry, SoCG 2019.*, 5, Leibniz International Proceedings in Informatics, LIPIcs, vol. 129, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 35th International Symposium on Computational Geometry, SoCG 2019, Portland, United States, 6/18/19. https://doi.org/10.4230/LIPIcs.SoCG.2019.5

}

TY - GEN

T1 - An efficient algorithm for generalized polynomial partitioning and its applications

AU - Agarwal, Pankaj K.

AU - Aronov, Boris

AU - Ezra, Esther

AU - Zahl, Joshua

PY - 2019/6/1

Y1 - 2019/6/1

N2 - In 2015, Guth proved that if S is a collection of n g-dimensional semi-algebraic sets in ℝd 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 ℝd \ Z(P) intersects O(n/Dd−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 Rd in O(log n) time, with storage complexity and expected preprocessing time of O(nd+ε). The second is a data structure for answering range search queries with semi-algebraic ranges in O(log n) time, with O(nt+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 ℝd in O(log2 n) time, with O(nd+ε) storage and expected preprocessing time. The fourth is an efficient algorithm for cutting algebraic planar curves into pseudo-segments.

AB - In 2015, Guth proved that if S is a collection of n g-dimensional semi-algebraic sets in ℝd 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 ℝd \ Z(P) intersects O(n/Dd−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 Rd in O(log n) time, with storage complexity and expected preprocessing time of O(nd+ε). The second is a data structure for answering range search queries with semi-algebraic ranges in O(log n) time, with O(nt+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 ℝd in O(log2 n) time, with O(nd+ε) storage and expected preprocessing time. The fourth is an efficient algorithm for cutting algebraic planar curves into pseudo-segments.

KW - Polynomial partitioning

KW - Quantifier elimination

KW - Semi-algebraic range spaces

KW - ε-samples

UR - http://www.scopus.com/inward/record.url?scp=85066829557&partnerID=8YFLogxK

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U2 - 10.4230/LIPIcs.SoCG.2019.5

DO - 10.4230/LIPIcs.SoCG.2019.5

M3 - Conference contribution

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - 35th International Symposium on Computational Geometry, SoCG 2019

A2 - Barequet, Gill

A2 - Wang, Yusu

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

ER -