Approximating Klee's Measure Problem and a Lower Bound for Union Volume Estimation

2024-10-01

arXiv.org

AI Summary

This paper studies the Klee’s measure problem: approximating the volume of the union of $n$ axis-aligned hyperrectangles in $d$-dimensional space under a restricted interaction model supporting only volume queries, uniform random sampling, and membership queries. The main contributions are threefold: (1) establishing the first tight $Omega(n/varepsilon^2)$ query complexity lower bound, proving the optimality of classical algorithms; (2) proposing the first subquadratic-time approximation algorithm, improving the time complexity from $O(n/varepsilon^2)$ to $Oig((n + 1/varepsilon^2) cdot log^{O(d)} nig)$ via geometric stratification—clustering rectangles by shape—and orthogonal range searching; and (3) demonstrating through experiments that the algorithm significantly outperforms existing baselines in dimensions 2–4.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Web measurements

Abstract

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.

Problem

• Estimating union volume of objects in constant dimension.
• Optimizing query complexity for Klee's measure problem.
• Improving algorithm efficiency using geometric properties.

Innovation

• Optimizes query complexity for union volume estimation
• Exploits geometric properties for efficient sampling
• Improves time complexity using orthogonal range searching