Union volume estimation is a classical algorithmic problem. Given a family of objects $O_1,\ldots,O_n \subseteq \mathbb{R}^d$, we want to approximate the volume of their union. In the special case where all objects are boxes (also known as hyperrectangles) this is known as Klee's measure problem. The state-of-the-art algorithm [Karp, Luby, Madras '89] for union volume estimation and Klee's measure problem in constant dimension $d$ computes a $(1+\varepsilon)$-approximation with constant success probability by using a total of $O(n/\varepsilon^2)$ queries of the form (i) ask for the volume of $O_i$, (ii) sample a point uniformly at random from $O_i$, and (iii) query whether a given point is contained in $O_i$. We show that if one can only interact with the objects via the aforementioned three queries, the query complexity of [Karp, Luby, Madras '89] is indeed optimal, i.e., $\Omega(n/\varepsilon^2)$ queries are necessary. Our lower bound already holds for estimating the union of equiponderous axis-aligned polygons in $\mathbb{R}^2$, and even if the algorithm is allowed to inspect the coordinates of the points sampled from the polygons, and still holds when a containment query can ask containment of an arbitrary (not sampled) point. Guided by the insights of the lower bound, we provide a more efficient approximation algorithm for Klee's measure problem improving the $O(n/\varepsilon^2)$ time to $O((n+\frac{1}{\varepsilon^2}) \cdot \log^{O(d)}n)$. We achieve this improvement by exploiting the geometry of Klee's measure problem in various ways: (1) Since we have access to the boxes' coordinates, we can split the boxes into classes of boxes of similar shape. (2) Within each class, we show how to sample from the union of all boxes, by using orthogonal range searching. And (3) we exploit that boxes of different classes have small intersection, for most pairs of classes.

翻译：并集体积估计是一个经典的算法问题。给定一族对象 $O_1,\ldots,O_n \subseteq \mathbb{R}^d$，我们希望近似其并集的体积。在所有对象均为盒子（亦称超矩形）的特殊情形下，该问题被称为Klee测度问题。针对常数维度 $d$ 下的并集体积估计与Klee测度问题，目前最优算法 [Karp, Luby, Madras '89] 通过使用总计 $O(n/\varepsilon^2)$ 次形式如下的查询：（i）询问 $O_i$ 的体积，（ii）从 $O_i$ 中均匀随机采样一个点，以及（iii）查询给定点是否包含于 $O_i$，以常数成功概率计算出 $(1+\varepsilon)$-近似。我们证明，若只能通过上述三种查询与对象交互，则 [Karp, Luby, Madras '89] 的查询复杂度确是最优的，即 $\Omega(n/\varepsilon^2)$ 次查询是必要的。我们的下界对于估计 $\mathbb{R}^2$ 中等权轴对齐多边形的并集已然成立，即使算法被允许检查从多边形中采样点的坐标，并且当包含查询可以询问任意（非采样）点的包含性时仍然成立。在下界见解的指导下，我们为Klee测度问题提供了一个更高效的近似算法，将 $O(n/\varepsilon^2)$ 的时间改进为 $O((n+\frac{1}{\varepsilon^2}) \cdot \log^{O(d)}n)$。我们通过多种方式利用Klee测度问题的几何特性实现这一改进：（1）由于我们可以访问盒子的坐标，我们可以将盒子划分为形状相似的类别。（2）在每个类别内，我们展示了如何通过使用正交范围搜索从所有盒子的并集中采样。（3）我们利用了对于大多数类别对，不同类别的盒子具有较小的交集。