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Dynamic and Streaming Algorithms for Union Volume Estimation

Sujoy Bhore, Karl Bringmann, Timothy M. Chan, Yanheng Wang

TL;DR

Algorithms for union volume estimation in the oracle model that support both insertions and deletions of objects and an algorithm supporting insertions and deletions in polylogarithmic update and query time and linear space are studied.

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.

Dynamic and Streaming Algorithms for Union Volume Estimation

TL;DR

Algorithms for union volume estimation in the oracle model that support both insertions and deletions of objects and an algorithm supporting insertions and deletions in polylogarithmic update and query time and linear space are studied.

Abstract

The union volume estimation problem asks to -approximate the volume of the union of given objects . In their seminal work in 1989, Karp, Luby, and Madras solved this problem in time in an oracle model where each object can be accessed via three types of queries: obtain the volume of , sample a random point from , and test whether contains a given point . 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.
Paper Structure (25 sections, 24 theorems, 20 equations, 1 figure, 6 algorithms)

This paper contains 25 sections, 24 theorems, 20 equations, 1 figure, 6 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Dynamic algorithm with insertions and deletions
  4. Streaming algorithm with insertions and suffix queries
  5. Streaming algorithm for convex bodies with insertions and deletions
  6. Technical Overview
  7. Dynamic algorithm with insertions and deletions
  8. Streaming algorithm with insertions and suffix queries
  9. Streaming algorithm for convex bodies with insertions and deletions
  10. Preliminaries
  11. Technical setup
  12. Reduction of volume range
  13. Volume estimation via sampling
  14. Dynamic Algorithm With Insertions and Deletions
  15. Preparations
  16. ...and 10 more sections

Key Result

Theorem 2.1

There is a dynamic algorithm in the oracle model that maintains a multiset $\mathcal{X}$ of objects under updates $X$ and $X$. Upon query , it outputs a $(1\pm\varepsilon)$-approximation to $\operatorname{vol}\left(\bigcup_{X \in \mathcal{X}} X\right)$ with high probability. The amortized expected u

Figures (1)

  • Figure 1: The convex body $X$, the inner John ellipsoid $E$, its inscribing box $G$, the subdivision into cells, and the minimum ellipsoid $F$ of the sample points.

Theorems & Definitions (44)

  • Theorem 2.1: Section \ref{['sec:dynamic']}
  • Theorem 2.2: Section \ref{['sec:window']}
  • Theorem 2.3: Section \ref{['sec:convex']}
  • Lemma 4.1
  • proof
  • Definition 4.2: $l$-sample
  • Definition 4.3: level
  • Lemma 4.4
  • proof
  • Theorem 5.1: Karp, Luby, Madras KarpLM89
  • ...and 34 more