Let $P$ be a set of $m$ points in ${\mathbb R}^2$, let $\Sigma$ be a set of $n$ semi-algebraic sets of constant complexity in ${\mathbb R}^2$, let $(S,+)$ be a semigroup, and let $w: P \rightarrow S$ be a weight function on the points of $P$. We describe a randomized algorithm for computing $w(P\cap\sigma)$ for every $\sigma\in\Sigma$ in overall expected time $O^*\bigl( m^{\frac{2s}{5s-4}}n^{\frac{5s-6}{5s-4}} + m^{2/3}n^{2/3} + m + n \bigr)$, where $s>0$ is a constant that bounds the maximum complexity of the regions of $\Sigma$, and where the $O^*(\cdot)$ notation hides subpolynomial factors. For $s\ge 3$, surprisingly, this bound is smaller than the best-known bound for answering $m$ such queries in an on-line manner. The latter takes $O^*(m^{\frac{s}{2s-1}}n^{\frac{2s-2}{2s-1}}+m+n)$ time. Let $\Phi: \Sigma \times P \rightarrow \{0,1\}$ be the Boolean predicate (of constant complexity) such that $\Phi(\sigma,p) = 1$ if $p\in\sigma$ and $0$ otherwise, and let $\Sigma\mathop{\Phi} P = \{ (\sigma,p) \in \Sigma\times P \mid \Phi(\sigma,p)=1\}$. Our algorithm actually computes a partition ${\mathcal B}_\Phi$ of $\Sigma\mathop{\Phi} P$ into bipartite cliques (bicliques) of size (i.e., sum of the sizes of the vertex sets of its bicliques) $O^*\bigl( m^{\frac{2s}{5s-4}}n^{\frac{5s-6}{5s-4}} + m^{2/3}n^{2/3} + m + n \bigr)$. It is straightforward to compute $w(P\cap\sigma)$ for all $\sigma\in \Sigma$ from ${\mathcal B}_\Phi$. Similarly, if $\eta: \Sigma \rightarrow S$ is a weight function on the regions of $\Sigma$, $\sum_{\sigma\in \Sigma: p \in \sigma} \eta(\sigma)$, for every point $p\in P$, can be computed from ${\mathcal B}_\Phi$ in a straightforward manner. A recent work of Chan et al. solves the online version of this dual point enclosure problem within the same performance bound as our off-line solution. We also mention a few other applications of computing ${\mathcal B}_\Phi$.

翻译：令$P$为${\mathbb R}^2$中$m$个点的集合，$\Sigma$为${\mathbb R}^2$中$n$个具有恒定复杂度的半代数集，$(S,+)$为一个半群，$w: P \rightarrow S$为$P$上点的权重函数。我们描述了一个随机算法，用于计算每个$\sigma\in\Sigma$对应的$w(P\cap\sigma)$，期望总时间为$O^*\bigl( m^{\frac{2s}{5s-4}}n^{\frac{5s-6}{5s-4}} + m^{2/3}n^{2/3} + m + n \bigr)$，其中$s>0$是控制$\Sigma$区域最大复杂度的常数，$O^*(\cdot)$记号隐藏了次多项式因子。令人惊讶的是，对于$s\ge 3$，该上界小于在线方式回答$m$个此类查询的最佳已知上界（后者需要$O^*(m^{\frac{s}{2s-1}}n^{\frac{2s-2}{2s-1}}+m+n)$时间）。设$\Phi: \Sigma \times P \rightarrow \{0,1\}$为（恒定复杂度的）布尔谓词，满足$\Phi(\sigma,p) = 1$当且仅当$p\in\sigma$，否则为$0$，并令$\Sigma\mathop{\Phi} P = \{ (\sigma,p) \in \Sigma\times P \mid \Phi(\sigma,p)=1\}$。我们的算法实际上计算了$\Sigma\mathop{\Phi} P$的一个划分${\mathcal B}_\Phi$，该划分由二部团（bicliques）构成，其规模（即所有二部团顶点集大小之和）为$O^*\bigl( m^{\frac{2s}{5s-4}}n^{\frac{5s-6}{5s-4}} + m^{2/3}n^{2/3} + m + n \bigr)$。从${\mathcal B}_\Phi$出发，可直接计算所有$\sigma\in \Sigma$对应的$w(P\cap\sigma)$。类似地，若$\eta: \Sigma \rightarrow S$为$\Sigma$区域上的权重函数，则对于每个点$p\in P$，$\sum_{\sigma\in \Sigma: p \in \sigma} \eta(\sigma)$也可从${\mathcal B}_\Phi$直接计算得出。Chan等人近期的工作以与我们离线解相同的性能界解决了该对偶点包含问题的在线版本。我们还提及了${\mathcal B}_\Phi$的若干其他应用。