Relative $(p,epsilon)$-Approximations in Geometry

We re-examine the notion of relative $(p,\eps)$-approximations, recently introduced in [CKMS06], and establish upper bounds on their size, in general range spaces of finite VC-dimension, using the sampling theory developed in [LLS01] and in several earlier studies [Pol86, Hau92, Tal94]. We also survey the different notions of sampling, used in computational geometry, learning, and other areas, and show how they relate to each other. We then give constructions of smaller-size relative $(p,\eps)$-approximations for range spaces that involve points and halfspaces in two and higher dimensions. The planar construction is based on a new structure–spanning trees with small relative crossing number, which we believe to be of independent interest. Relative $(p,\eps)$-approximations arise in several geometric problems, such as approximate range counting, and we apply our new structures to obtain efficient solutions for approximate range counting in three dimensions. We also present a simple solution for the planar case.

💡 Research Summary

The paper revisits the concept of relative (p, ε)-approximations, originally introduced for geometric range spaces, and establishes new size bounds for such approximations in settings of finite VC‑dimension. By relating relative (p, ε)-approximations to the (ν, α)-sampling framework of Li, Long, and Srinivasan (LLS01), the authors show that choosing ν proportional to p and α proportional to ε makes the two notions equivalent (Theorem 2.9). Consequently, the powerful sampling result of LLS01 yields relative (p, ε)-approximations of size O(δ · ε⁻² · p⁻¹ · log (1/p)) with constant probability, improving over the classic VC‑bound O(δ · ε⁻² · log (1/ε · p⁻¹)) by a factor of roughly 1/p.

The paper then focuses on two concrete geometric scenarios. In the planar case (points and half‑plane ranges) the authors construct a novel “spanning tree with low relative crossing number”. For any integer k ≤ n/2, any k‑shallow line (a line that leaves at most k points on one side) crosses at most O(√k · log (n/k)) edges of the tree. This refines the classic Welzl construction, which only guarantees O(√n) crossings for the deepest line. By sampling the vertices of such a tree, they obtain relative (p, ε)-approximations of size O(ε⁻⁴⁄³ · p⁻¹ · log (1/(εp))) and, simultaneously, sensitive ε‑approximations with an improved error term: |X(r) − Z(r)| ≤ (½) ε³⁄² · X(r)¹⁄⁴ + ε², which is tighter when X(r) > ε².

For higher dimensions, especially ℝ³, the spanning‑tree technique does not extend directly. Instead, the authors rely on Matoušek’s shallow‑partition theorem. They recursively build a sequence of approximation sets, each tailored to ranges containing roughly p·n, 2p·n, 4p·n, … points. Each set has size O(ε⁻³⁄² · p⁻¹ · log (1/(εp))) and provides an absolute error of at most εp for ranges of the corresponding size. Because the sizes decrease geometrically, the total size is dominated by the first set, yielding an overall relative (p, ε)-approximation of the same order. In dimensions d ≥ 3 the method can be generalized using higher‑dimensional shallow partitions, though additional constraints between ε and p (e.g., ε = O(p^{1/d})) appear.

Finally, the paper integrates these relative approximations into standard range‑search data structures. In the plane, the resulting structure answers ε‑approximate half‑plane counting queries in O(log n) time with linear space. In three dimensions, the authors achieve query times essentially polylogarithmic, a substantial improvement over previous algorithms that required O(n^{1‑1/⌊d/2⌋}) query time for exact counting. The techniques also extend to general semi‑algebraic ranges.

Overall, the work bridges learning‑theoretic sampling results with geometric data‑structure design, delivering tighter bounds for relative (p, ε)-approximations and demonstrating their practical impact on approximate range counting in both planar and three‑dimensional settings.