Abstract
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.
| Original language | English (US) |
|---|---|
| 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 |
|---|---|
| Volume | 129 |
| ISSN (Print) | 1868-8969 |
Conference
| Conference | 35th International Symposium on Computational Geometry, SoCG 2019 |
|---|---|
| Country | United States |
| City | Portland |
| Period | 6/18/19 → 6/21/19 |
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Keywords
- Polynomial partitioning
- Quantifier elimination
- Semi-algebraic range spaces
- ε-samples
ASJC Scopus subject areas
- Software
Cite this
An efficient algorithm for generalized polynomial partitioning and its applications. / Agarwal, Pankaj K.; Aronov, Boris; Ezra, Esther; Zahl, Joshua.
35th International Symposium on Computational Geometry, SoCG 2019. ed. / Gill Barequet; Yusu Wang. Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, 2019. 5 (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 129).Research output: Chapter in Book/Report/Conference proceeding › Conference contribution
}
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
UR - http://www.scopus.com/inward/citedby.url?scp=85066829557&partnerID=8YFLogxK
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 -