SIAM Journal on Computing, Volume 49, Issue 6, Page 1109-1127, January 2020.
In 2015, Guth [Math. Proc. Cambridge Philos. Soc., 159 (2015), pp. 459--469] proved that for any set of $k$-dimensional bounded complexity varieties in ${\mathbb R}^d$ and for any positive integer $D$, there exists a polynomial of degree at most $D$ whose zero set divides ${\mathbb R}^d$ into open connected sets so that only a small fraction of the given varieties intersect each of these sets. Guth's result generalized an earlier result of Guth and Katz [Ann. Math., 181 (2015), pp. 155--190] for points. Guth's proof relies on a variant of the Borsuk--Ulam theorem, and for $k>0$, it is unknown how to obtain an explicit representation of such a partitioning polynomial and how to construct it efficiently. In particular, it is unknown how to effectively construct such a polynomial for bounded-degree algebraic curves (or even lines) in ${{\mathbb R}}^3$. We present an efficient algorithmic construction for this setting. Given a set of $n$ input algebraic curves and a positive integer $D$, we efficiently construct a decomposition of space into $O(D^3\log^3{D})$ open “cells,” each of which meets $O(n/D^2)$ curves from the input. The construction time is $O(n^2)$. For the case of lines in 3-space, we present an improved implementation whose running time is $O(n^{4/3} { polylog }{n})$. The constant of proportionality in both time bounds depends on $D$ and the maximum degree of the polynomials defining the input curves. As an application, we revisit the problem of eliminating depth cycles among nonvertical lines in 3-space, recently studied by Aronov and Sharir [Discrete Comput. Geom., 59 (2018), pp. 725--741] and show an algorithm that cuts $n$ such lines into $O(n^{3/2+\varepsilon})$ pieces that are depth-cycle free for any $\varepsilon > 0$. The algorithm runs in $O(n^{3/2+\varepsilon})$ time, which is a considerable improvement over the previously known algorithms.

中文翻译：

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$ \ mathbb {R} ^ 3 $中代数曲线的构造多项式划分及其应用

SIAM计算杂志，第49卷，第6期，第1109-1127页，2020年1月。
2015年，古斯[数学。进程 剑桥Philos。[Soc。，159（2015），pp。459--469]证明，对于$ {\ mathbb R} ^ d $中的任何$ k $维有界复杂性集合和任何正整数$ D $，都存在一个最多为$ D $的度数多项式，其零集将$ {\ mathbb R} ^ d $分成开放的连通集，这样，给定变体中只有一小部分与这些集合中的每一个相交。Guth的结果推广了Guth和Katz [Ann。Math。181（2015），第155--190页]。Guth的证明依赖于Borsuk-Ulam定理的一个变体，对于$ k> 0 $，未知如何获得这种划分多项式的显式表示以及如何有效地构造它。尤其是，对于$ {{\ mathbb R}} ^ 3 $中的有界代数曲线（或什至线），如何有效构造这样的多项式是未知的。我们为该设置提出了一种有效的算法构造。给定一组$ n $输入代数曲线和一个正整数$ D $，我们有效地将空间分解为$ O（D ^ 3 \ log ^ 3 {D}）$个开放的“单元”，每个单元都满足输入的$ O（n / D ^ 2）$曲线。构造时间为$ O（n ^ 2）$。对于3空间中的行，我们提出一种改进的实现，其运行时间为$ O（n ^ {4/3} {polylog} {n}）$。两个时间范围内的比例常数取决于$ D $和定义输入曲线的多项式的最大次数。作为一种应用，我们将重新讨论消除3空间中非垂直线之间的深度循环的问题，最近由Aronov和Sharir [Discrete Comput。Geom。，59（2018），pp。725--741]，并展示了一种算法，该算法可将$ n $此类行切成$ O（n ^ {3/2 + \ varepsilon}）$块，这些块对于深度循环是免费的任何$ \ varepsilon> 0 $。该算法以$ O（n ^ {3/2 + \ varepsilon}）$的时间运行，这是对先前已知算法的显着改进。