半代数值域搜索和Stabbing问题的下界
Lower Bounds for Semialgebraic Range Searching and Stabbing Problems
摘要：
在半代数范围搜索问题中，我们要对$\mathbb{R}^d$ s. t中的$n$点进行预处理.对于来自常复杂度半代数集合族的任何查询范围，可以有效地报告或计数与该范围相交的所有点。当范围由单纯形组成时，可以使用$S(n)$空间和使用$S(n)Q^d(n) = \tilde{O}(n^d)$的$Q(n)$查询时间来解决问题，并且这种权衡几乎是紧密的。因此，存在使用具有$O(n^{1-1/d})$查询时间的$\tilde{O}(n)$空间的低空间结构和使用具有$O(\log^{d} n)$查询时间的$O(n^d)$空间的快速查询结构。然而，对于一般的半代数范围，只有低空间的解决方案是已知的，但最好的解决方案匹配相同的权衡曲线作为单纯形查询。已经证实，对于快速查询的情况也可以这样做，但是这个开放的问题仍然没有解决。在这里，我们反驳这个猜想。给出了半极大值域搜索的第一非平凡下界及相关问题。我们表明，任何数据结构报告两个同心圆之间的点与$Q(n)$查询时间必须使用$S(n)=Ω(n^{3-o(1)}/Q(n)^5)$空间，这意味着，对于$Q(n)=O(\log^{O(1)}n)$，$Ω(n^{3-o(1)})$空间必须使用。我们还研究了两个多项式的形式$Y=\sum_{i=0}^Δa_i X^i$，其中$a_0, \cdots, a_Δ$在查询时给出的报告点之间的问题。我们展示了$S(n)=Ω(n^{Δ+1-o(1)}/Q(n)^{Δ^2+Δ})$。所以对于$Q(n)=O(\log^{O(1)}n)$，我们必须使用$Ω(n^{Δ+1-o(1)})$空间。对于对偶半代数刺伤问题，我们表明，在线性空间中，任何数据结构，解决2D环刺伤必须使用$Ω(n^{2/3})$查询时间。这几乎与线性化上限相匹配。对于一般的半代数板刺问题，再次，我们显示了一个几乎紧的下界。
英文摘要：
In the semialgebraic range searching problem, we are to preprocess $n$ points in $\mathbb{R}^d$ s.t. for any query range from a family of constant complexity semialgebraic sets, all the points intersecting the range can be reported or counted efficiently. When the ranges are composed of simplices, the problem can be solved using $S(n)$ space and with $Q(n)$ query time with $S(n)Q^d(n) = \tilde{O}(n^d)$ and this trade-off is almost tight. Consequently, there exists low space structures that use $\tilde{O}(n)$ space with $O(n^{1-1/d})$ query time and fast query structures that use $O(n^d)$ space with $O(\log^{d} n)$ query time. However, for the general semialgebraic ranges, only low space solutions are known, but the best solutions match the same trade-off curve as the simplex queries. It has been conjectured that the same could be done for the fast query case but this open problem has stayed unresolved. Here, we disprove this conjecture. We give the first nontrivial lower bounds for semilagebraic range searching and related problems. We show that any data structure for reporting the points between two concentric circles with $Q(n)$ query time must use $S(n)=Ω(n^{3-o(1)}/Q(n)^5)$ space, meaning, for $Q(n)=O(\log^{O(1)}n)$, $Ω(n^{3-o(1)})$ space must be used. We also study the problem of reporting the points between two polynomials of form $Y=\sum_{i=0}^Δa_i X^i$ where $a_0, \cdots, a_Δ$ are given at the query time. We show $S(n)=Ω(n^{Δ+1-o(1)}/Q(n)^{Δ^2+Δ})$. So for $Q(n)=O(\log^{O(1)}n)$, we must use $Ω(n^{Δ+1-o(1)})$ space. For the dual semialgebraic stabbing problems, we show that in linear space, any data structure that solves 2D ring stabbing must use $Ω(n^{2/3})$ query time. This almost matches the linearization upper bound. For general semialgebraic slab stabbing problems, again, we show an almost tight lower bounds.
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