Relative (p,ε)-Approximations in Geometry

Abstract

We re-examine the notion of relative (p, ε)-approximations, recently introduced in Cohen et al. (Manuscript, 2006), and establish upper bounds on their size, in general range spaces of finite VC-dimension, using the sampling theory developed in Li et al. (J. Comput. Syst. Sci. 62:516-527, 2001) and in several earlier studies (Pollard in Manuscript, 1986; Haussler in Inf. Comput. 100:78-150, 1992; Talagrand in Ann. Probab. 22:28-76, 1994). 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, ε)-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, ε)-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.

A Size-Sensitive Discrepancy Bound for Set Systems of Bounded Primal Shatter Dimension

“…Our bound improves and generalizes the bounds obtained from the machinery of Har-Peled and Sharir [14] (and the follow-up work in [26]) for points and halfspaces in d-space for d ≥ 3.…”

“…As observed by Har-Peled and Sharir [14], relative (ε, δ)-approximations and (ν, α)-samples are equivalent with an appropriate relation between ε, δ, and ν, α (roughly speaking, they are equivalent up to some constant factor). Due to this observation they conclude that the analysis of Li et al [18] (that shows a bound on the size of (ν, α)-samples) implies that for set systems of finite VC-dimension d, there exist relative (ε, δ)-approximations of size…”

“…The motivation to establish sensitive discrepancy bounds of the above kind stems from their application in the construction of relative (ε, δ)-approximations, introduced by Har-Peled and Sharir [14] 1 based on the work of Li et al [18] in the context of machine learning theory. We recall the definition from [14]: For a set system (X, S) (with X finite), the measure of a set S ∈ S is the quantity X(S) = |S∩X| |X| .…”

“…The motivation to establish sensitive discrepancy bounds of the above kind stems from their application in the construction of relative (ε, δ)-approximations, introduced by Har-Peled and Sharir [14] 1 based on the work of Li et al [18] in the context of machine learning theory. We recall the definition from [14]: For a set system (X, S) (with X finite), the measure of a set S ∈ S is the quantity X(S) = |S∩X| |X| . Given a set system (X, S) and two parameters, 0 < ε < 1 and 0 < δ < 1, we say that a subset Z ⊆ X is a relative (ε, δ)-approximation if it satisfies, for each set S ∈ S, X(S)(1 − δ) ≤ Z(S) ≤ X(S)(1 + δ), if X(S) ≥ ε, and X(S) − δε ≤ Z(S) ≤ X(S) + δε, otherwise.…”

Adaptive and Approximate Orthogonal Range Counting

“…The idea of sampling has been used in previous approximate range counting papers [2,20]. Consider a random sample R ⊆ P where each point is chosen independently with probability r/n.…”