SIAM Journal on Computing, Volume 50, Issue 2, Page 760-787, January 2021.
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 $P$ of degree at most $D$ so that each connected component of $\mathbb{R}^d\setminus Z(P)$ intersects $O(n/D^{d-g})$ sets from $\EuScript{S}$. 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 $\lvert \EuScript{S}\rvert$. Our approach exploits the technique of quantifier elimination combined with that of $\eps$-samples. We also present an extension of our construction to multilevel polynomial partitioning for semialgebraic sets in $\mathbb{R}^d$. We present five applications of our result. The first is a data structure for answering point-enclosure queries among a family of semialgebraic sets in $\mathbb{R}^d$ in $O(\log n)$ time, with storage complexity and expected preprocessing time of $O(n^{d+\eps})$. The second is a data structure for answering range-searching queries with semialgebraic ranges in $\mathbb{R}^d$ in $O(\log n)$ time, with $O(n^{t+\eps})$ 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 semialgebraic sets in $\mathbb{R}^{d}$ in $O(\log^2 n)$ time, with $O(n^{d+\eps})$ storage and expected preprocessing time. The fourth is an efficient algorithm for cutting algebraic curves in $\mathbb{R}^2$ into pseudosegments. The fifth application is for eliminating depth cycles among triangles in $\mathbb{R}^3$, where we show a nearly optimal algorithm to cut $n$ pairwise disjoint nonvertical triangles in ${\mathbb{R}}^3$ into pieces that form a depth order.

中文翻译：

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广义多项式划分的高效算法及其应用

SIAM Journal on Computing，第 50 卷，第 2 期，第 760-787 页，2021 年 1 月。
2015 年，Guth 证明了如果 $\EuScript{S}$ 是 ${\mathbb{R}}^d$ 中 $n$ $g$ 维半代数集的集合，并且如果 $D\geq 1$ 是整数，则存在一个最多为 $D$ 的 $d$-变量多项式 $P$，使得 $\mathbb{R}^d\setminus Z(P)$ 的每个连通分量都与 $O(n/D ^{dg})$ 从 $\EuScript{S}$ 设置。这样的多项式称为广义划分多项式。我们提出了一种高效计算此类多项式的随机算法——我们算法的预期运行时间在 $\lvert \EuScript{S}\rvert$ 中是线性的。我们的方法利用了量词消除技术与 $\eps$-samples 相结合的技术。我们还在 $\mathbb{R}^d$ 中展示了我们的构造扩展到半代数集的多级多项式划分。我们展示了我们结果的五种应用。第一个是一个数据结构，用于在 $O(\log n)$ 时间内回答 $\mathbb{R}^d$ 中的半代数集族中的点-包围查询，其存储复杂度和预期的预处理时间为 $O( n^{d+\eps})$。第二个是在 $O(\log n)$ 时间内用 $\mathbb{R}^d$ 中的半代数范围回答范围搜索查询的数据结构，具有 $O(n^{t+\eps})$ 存储和预期的预处理时间，其中 $t > 0$ 是一个取决于 $d$ 和范围描述复杂度的整数。第三个是在 $O(\log^2 n)$ 时间内回答 $\mathbb{R}^{d}$ 中半代数集合之间的垂直射线射击查询的数据结构，其中 $O(n^{d+\ eps})$ 存储和预期的预处理时间。第四个是将 $\mathbb{R}^2$ 中的代数曲线切割成伪段的有效算法。