The union volume estimation problem asks to $(1\pm\varepsilon)$-approximate the volume of the union of $n$ given objects $X_1,\ldots,X_n \subset \mathbb{R}^d$. In their seminal work in 1989, Karp, Luby, and Madras solved this problem in time $O(n/\varepsilon^2)$ in an oracle model where each object $X_i$ can be accessed via three types of queries: obtain the volume of $X_i$, sample a random point from $X_i$, and test whether $X_i$ contains a given point $x$. This running time was recently shown to be optimal [Bringmann, Larsen, Nusser, Rotenberg, and Wang, SoCG'25]. In another line of work, Meel, Vinodchandran, and Chakraborty [PODS'21] designed algorithms that read the objects in one pass using polylogarithmic time per object and polylogarithmic space; this can be phrased as a dynamic algorithm supporting insertions of objects for union volume estimation in the oracle model. In this paper, we study algorithms for union volume estimation in the oracle model that support both insertions and deletions of objects. We obtain the following results: - an algorithm supporting insertions and deletions in polylogarithmic update and query time and linear space (this is the first such dynamic algorithm, even for 2D triangles); - an algorithm supporting insertions and suffix queries (which generalizes the sliding window setting) in polylogarithmic update and query time and space; - an algorithm supporting insertions and deletions of convex bodies of constant dimension in polylogarithmic update and query time and space.

翻译：并集体积估计问题要求以$(1\pm\varepsilon)$的精度近似计算给定$n$个对象$X_1,\ldots,X_n \subset \mathbb{R}^d$的并集体积。在1989年的开创性工作中，Karp、Luby和Madras在预言机模型中以$O(n/\varepsilon^2)$的时间复杂度解决了该问题，其中每个对象$X_i$可通过三类查询进行访问：获取$X_i$的体积、从$X_i$中采样随机点、以及检测$X_i$是否包含给定点$x$。近期研究证明该时间复杂度是最优的[Bringmann, Larsen, Nusser, Rotenberg, and Wang, SoCG'25]。在另一系列工作中，Meel、Vinodchandran和Chakraborty[PODS'21]设计了单遍读取对象的算法，每个对象处理时间为多对数级且空间复杂度为多对数级；这可以表述为预言机模型中支持对象插入操作的动态并集体积估计算法。本文研究预言机模型中支持对象插入与删除操作的并集体积估计算法，获得以下成果：- 支持插入与删除操作的算法，具有多对数级更新/查询时间与线性空间复杂度（这是首个此类动态算法，即使对于二维三角形情形）；- 支持插入操作与后缀查询（可推广至滑动窗口场景）的算法，具有多对数级更新/查询时间与空间复杂度；- 支持常数维凸体插入与删除操作的算法，具有多对数级更新/查询时间与空间复杂度。