arXiv 2203.070962022-03-15cs.CG

关于半代数范围报告

On Semialgebraic Range Reporting

Peyman Afshani, Pingan Cheng

备注：: Full version of the SoCG'22 paper

摘要：

在半代数范围搜索问题中，我们要对$\mathbb{R}^D$中的一组点进行预处理，使得可以有效地找到由$O(1)$次多项式不等式描述的半代数区域内的点的子集.最近，在这个问题上取得了一些重大进展。利用代数技巧，得到了$Q(n)=O(n^{1-1/D+o(1)})$具有几乎最优查询时间的“近线性空间”结构[AMS 13，MP 15]。对于“快速查询”结构（即，当$Q(n)=n^{o(1)}$）时，证实了具有空间$S(n) = O(n^{D+o(1)})$的结构是可能的。这个猜想最近被Afshani和Cheng反驳了[AC 21]。在平面上，他们证明了$Q(n) = n^{o(1)}$需要表示$Ω(n^{Δ+1-o(1)})$空间的$S(n) = Ω(n^{Δ+1 - o(1)}/Q(n)^{(Δ+3)Δ/2})$。虽然这反驳了猜想，但它仍然留下了一些未解决的问题：下界仅适用于2D和快速查询，并且即使对于$D=2$，$n$或$Q(n)$的指数似乎也不紧，因为当前的上限是$S(n) = O(n^{\boldsymbol{m}+o(1)}/Q(n)^{(\boldsymbol{m}-1)D/(D-1)})$，其中$\boldsymbol{m}=\binom{D+Δ}{D}-1 = Ω(Δ^D)$是定义一元度$Δ$ $D$变量多项式的最大参数数，对于任何$D,Δ=O(1)$。在本文中，我们解决了两个问题：我们证明了一个下界$D$ -维，并表明，当$Q(n)=n^{o(1)}+O(k)$，$S(n)=Ω(n^{\boldsymbol{m}-o(1)})$，这是几乎紧的$n$的指数被认为是在指针机器模型。当考虑$Q(n)$的指数时，我们通过给出均匀随机点集的匹配上界来证明[AC 21]中的分析对于$D=2$是紧的。这表明现有的上界可以改进，或者需要一个新的根本不同的输入集来获得更好的下界。

英文摘要：

In the problem of semialgebraic range searching, we are to preprocess a set of points in $\mathbb{R}^D$ such that the subset of points inside a semialgebraic region described by $O(1)$ polynomial inequalities of degree $Δ$ can be found efficiently. Relatively recently, several major advances were made on this problem. Using algebraic techniques, "near-linear space" structures [AMS13,MP15] with almost optimal query time of $Q(n)=O(n^{1-1/D+o(1)})$ were obtained. For "fast query" structures (i.e., when $Q(n)=n^{o(1)}$), it was conjectured that a structure with space $S(n) = O(n^{D+o(1)})$ is possible. The conjecture was refuted recently by Afshani and Cheng [AC21]. In the plane, they proved that $S(n) = Ω(n^{Δ+1 - o(1)}/Q(n)^{(Δ+3)Δ/2})$ which shows $Ω(n^{Δ+1-o(1)})$ space is needed for $Q(n) = n^{o(1)}$. While this refutes the conjecture, it still leaves a number of unresolved issues: the lower bound only works in 2D and for fast queries, and neither the exponent of $n$ or $Q(n)$ seem to be tight even for $D=2$, as the current upper bound is $S(n) = O(n^{\boldsymbol{m}+o(1)}/Q(n)^{(\boldsymbol{m}-1)D/(D-1)})$ where $\boldsymbol{m}=\binom{D+Δ}{D}-1 = Ω(Δ^D)$ is the maximum number of parameters to define a monic degree-$Δ$ $D$-variate polynomial, for any $D,Δ=O(1)$. In this paper, we resolve two of the issues: we prove a lower bound in $D$-dimensions and show that when $Q(n)=n^{o(1)}+O(k)$, $S(n)=Ω(n^{\boldsymbol{m}-o(1)})$, which is almost tight as far as the exponent of $n$ is considered in the pointer machine model. When considering the exponent of $Q(n)$, we show that the analysis in [AC21] is tight for $D=2$, by presenting matching upper bounds for uniform random point sets. This shows either the existing upper bounds can be improved or a new fundamentally different input set is needed to get a better lower bound.

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