动态范围计数的单元探测复杂度
The Cell Probe Complexity of Dynamic Range Counting
摘要：
在本文中，我们开发了一种新的技术，用于证明细胞探针模型中的动态数据结构的更新时间和查询时间的下界。利用这种技术，我们证明了最高的下限，以日期为任何明确的问题，即下限的$t_q=Ω((\lg n/\lg(wt_u))^2)$。这里$n$是更新操作的次数，$w$是单元大小，$t_q$是查询时间，$t_u$是更新时间。在单元大小$w=Θ(\lg n)$的最自然设置中，这给出了任何多对数更新时间的$t_q=Ω((\lg n/\lg \lg n)^2)$的下限。该界限几乎是$Ω(\lg n)$之前最高下限的二次改进，这是由于Pagetrachcu和Demaine [SICOMP'06]。我们证明了加权正交范围计数基本问题的下界。在这个问题中，我们要支持二维点的插入，每个点都被分配了一个$Θ(\lg n)$位整数权重。对该问题的查询由点$q=(x,y)$指定，并且目标是报告分配给由$q$支配的点的权重的总和，其中如果$x' \leq x$和$y' \leq y$，则点$(x',y')$由$q$支配。除了是迄今为止最高的单元探测下限之外，对于具有更新时间$t_u = Ω(\lg^{2+\eps}n)$的数据结构，该下限也是紧的，其中$\eps>0$是任意小的常数。
英文摘要：
In this paper we develop a new technique for proving lower bounds on the update time and query time of dynamic data structures in the cell probe model. With this technique, we prove the highest lower bound to date for any explicit problem, namely a lower bound of $t_q=Ω((\lg n/\lg(wt_u))^2)$. Here $n$ is the number of update operations, $w$ the cell size, $t_q$ the query time and $t_u$ the update time. In the most natural setting of cell size $w=Θ(\lg n)$, this gives a lower bound of $t_q=Ω((\lg n/\lg \lg n)^2)$ for any polylogarithmic update time. This bound is almost a quadratic improvement over the highest previous lower bound of $Ω(\lg n)$, due to Pǎtraşcu and Demaine [SICOMP'06]. We prove the lower bound for the fundamental problem of weighted orthogonal range counting. In this problem, we are to support insertions of two-dimensional points, each assigned a $Θ(\lg n)$-bit integer weight. A query to this problem is specified by a point $q=(x,y)$, and the goal is to report the sum of the weights assigned to the points dominated by $q$, where a point $(x',y')$ is dominated by $q$ if $x' \leq x$ and $y' \leq y$. In addition to being the highest cell probe lower bound to date, the lower bound is also tight for data structures with update time $t_u = Ω(\lg^{2+\eps}n)$, where $\eps>0$ is an arbitrarily small constant.
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