2024/10/01 by Karl Bringmann, Bringmann, Karl, Kasper Green Larsen +7 · 1 voice
Computer Science · #Computational Complexity (cs.CC) #Computational Geometry (cs.CG) #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #cs.CC #cs.CG #cs.DS
paper · pdf · doi:10.48550/arxiv.2410.00996
arxiv published 2024/10/01 · arxiv updated 2025/03/25
Union volume estimation is a classical algorithmic problem. Given a family of objects O1,…,On ⊆ ℝd, 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+ε)-approximation with constant success probability by using a total of O(n/ε2) queries of the form (i) ask for the volume of Oi, (ii) sample a point uniformly at random from Oi, and (iii) query whether a given point is contained in Oi. 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., Ω(n/ε2) queries are necessary. Our lower bound already holds for estimating the union of equiponderous axis-aligned polygons in ℝ2, 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/ε2) time to O((n+(1)/(ε2)) ⋅ logO(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.