代数决策树模型中若干\textsc{3Sum} -难几何问题的次二次算法
Subquadratic Algorithms for Some \textsc{3Sum}-Hard Geometric Problems in the Algebraic Decision Tree Model
摘要：
我们在代数决策树模型中提出了几个\textsc{3Sum}困难几何问题的次二次算法，所有这些问题都可以归结为以下问题：给定两个集合$A$、$B$，每个集合由平面中的$n$成对不相交的线段组成，以及平面中的$n$三角形的集合$C$，我们想要对每个三角形$Δ\in C$计数，$A$的片段与位于$Δ$中的$B$的片段之间的交点的数量。Chan~（2020）已经研究了本文中考虑的问题，他给出了在标准实RAM模型中在$O((n^2/\log^2n)\log^{O(1)}\log n)$时间内解决这些问题的算法。我们提出的解决方案，在代数决策树模型的成本是$O(n^{60/31+\varepsilon})$，任何$\varepsilon>0$。我们的方法基于原始-对偶范围搜索机制，该机制利用了Agarwal，Aronov，Ezra和Zahl~（2020）最近开发的多级多项式划分机制。在该过程中的一个关键步骤是一个变量的点位置的安排，说的线在平面上，这是完全基于\emph{order type}的线，一个“障碍”，原来是有利于加快我们的算法。
英文摘要：
We present subquadratic algorithms in the algebraic decision-tree model for several \textsc{3Sum}-hard geometric problems, all of which can be reduced to the following question: Given two sets $A$, $B$, each consisting of $n$ pairwise disjoint segments in the plane, and a set $C$ of $n$ triangles in the plane, we want to count, for each triangle $Δ\in C$, the number of intersection points between the segments of $A$ and those of $B$ that lie in $Δ$. The problems considered in this paper have been studied by Chan~(2020), who gave algorithms that solve them, in the standard real-RAM model, in $O((n^2/\log^2n)\log^{O(1)}\log n)$ time. We present solutions in the algebraic decision-tree model whose cost is $O(n^{60/31+\varepsilon})$, for any $\varepsilon>0$. Our approach is based on a primal-dual range searching mechanism, which exploits the multi-level polynomial partitioning machinery recently developed by Agarwal, Aronov, Ezra, and Zahl~(2020). A key step in the procedure is a variant of point location in arrangements, say of lines in the plane, which is based solely on the \emph{order type} of the lines, a "handicap" that turns out to be beneficial for speeding up our algorithm.
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