On range searching with semialgebraic sets II

Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F13%3A10173445" target="_blank" >RIV/00216208:11320/13:10173445 - isvavai.cz</a>
Result on the web
<a href="http://arxiv.org/abs/1208.3384" target="_blank" >http://arxiv.org/abs/1208.3384</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.1137/120890855" target="_blank" >10.1137/120890855</a>

Result language
angličtina
Original language name
On range searching with semialgebraic sets II
Original language description
Let $P$ be a set of $n$ points in $R^d$. We present a linear-size data structure for answering range queries on $P$ with constant-complexity semialgebraic sets as ranges, in time close to $O(n^{1-1/d})$. It essentially matches the performance of similarstructures for simplex range searching, and, for $dge 5$, significantly improves earlier solutions by the first two authors obtained in~1994. This almost settles a long-standing open problem in range searching. The data structure is based on the polynomial-partitioning technique of Guth and Katz, which shows that for a parameter $r$, $1 < r le n$, there exists a $d$-variate polynomial $f$ of degree $O(r^{1/d})$ such that each connected component of $R^dsetminus Z(f)$ contains at most $n/r$ points of $P$, where $Z(f)$ is the zero set of $f$. We present an efficient randomized algorithm for computing such a polynomial partition, which is of independent interest and is likely to have additional applications.
Czech name
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Czech description
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Type
J<sub>x</sub> - Unclassified - Peer-reviewed scientific article (Jimp, Jsc and Jost)
CEP classification
IN - Informatics
OECD FORD branch
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Project
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Continuities
I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year
2013
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

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