On Range Searching with Semialgebraic Sets II
Abstract
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 similar structures for simplex range searching, and, for
d≥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 [arXiv:1011.4105], which shows that for a parameter
r
,
1<r≤n
, there exists a
d
-variate polynomial
f
of degree
O(
r
1/d
)
such that each connected component of $\R^d\setminus Z(f)$ 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.