Abstract
In 2015, Guth proved that if $\EuScript{S}$ is a collection of $n$ $g$-dimensional semialgebraic sets in ${\mathbb{R}}^d$ and if $D\geq 1$ is an integer, then there is a $d$-variate polynomial $P$ ...
Highlights
In 2015, Guth [11] proved that if S is a collection of n g-dimensional semi-algebraic sets1 in Rd and if D ≥ 1 is an integer, 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 2, where the implicit constant in the O(·) notation depends on d and on the degree and number of polynomials required to define each semi-algebraic set
An efficient algorithm for computing a generalized partitioning polynomial leads to a semialgebraic range searching data structure with O(log n) query time and O(nt+ε) space
We restrict each of the polynomials P1, . . . , Pk to S and apply Proposition 3 on this restricted collection, thereby obtaining a set of points meeting each connected component of each of the realizable sign conditions, as well as the corresponding list of signs of the restricted polynomials for each of these points
Summary
In 2015, Guth [11] proved that if S is a collection of n g-dimensional semi-algebraic sets in Rd and if D ≥ 1 is an integer, 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 2, where the implicit constant in the O(·) notation depends on d and on the degree and number of polynomials required to define each semi-algebraic set. In [2], Agarwal, Matoušek, and Sharir developed an efficient algorithm to compute partitioning polynomials, matching the degree bound obtained in [12] up to a constant factor They used this algorithm to construct a linear-size data structure that can answer semialgebraic range queries amid a set of n points in Rd in time O(n1−1/d polylog(n)), which is near optimal. An efficient algorithm for computing a generalized partitioning polynomial leads to a semialgebraic range searching data structure with O(log n) query time and O(nt+ε) space. We present a data structure of size O(nd+ε), for any constant ε > 0, that can compute, in O(log n) time, the cumulative weight of the sets in S containing a query point. Whenever big-O notation is used, the implicit constant may depend on d, b, and ε
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