Dynamic and Streaming Algorithms for Union Volume Estimation

2026-02-18

AI Summary

This work addresses the problem of estimating the volume of the union of geometric objects under dynamic insertions and deletions. In the oracle model, it presents the first approximation algorithm supporting dynamic updates, achieving a $(1 \pm \varepsilon)$-approximation for both queries and updates with high efficiency. The approach applies to two-dimensional triangles and constant-dimensional convex bodies, and further extends to suffix-based query models such as sliding windows. By leveraging three types of oracle operations—volume queries, point sampling, and membership testing—the proposed method designs dynamic data structures and streaming algorithms that achieve polylogarithmic update and query times for a broad class of objects, while using only linear space. This significantly enhances computational efficiency in dynamic and streaming settings.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: 3D Computer Vision
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

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.

Problem

• union volume estimation
• dynamic algorithms
• streaming algorithms
• insertions and deletions
• oracle model

Innovation

• dynamic algorithms
• union volume estimation
• streaming algorithms
• convex bodies
• oracle model