Testing Surface Area
Pravesh K. Kothari, Amir Nayyeri, Chenggang Wu · 2013
We consider the problem of estimating the surface area of an unknown n-dimensional set F given membership oracle access. In contrast to previous work, we do not assume that F is convex, and in fact make no assumptions at all about F. By necessity this means that we work in the property testing model; we seek an algorithm which, given parameters A and ϵ, satisfies: • if surf(F) ≤ A then the algorithm accepts (whp); • if F is not ϵ-close to some set G with surf(G) ≤ κA, then the algorithm rejects (whp). We call κ ≥ 1 the “approximation factor ” of the testing algorithm. The n = 1 case (in which “surf(F) = 2m ” means F is a disjoint union of m intervals) was introduced by Kearns and Ron [KR98], who solved the problem with κ = 1/ϵ and O(1/ϵ) oracle queries. Later, Balcan et al. [BBBY12] solved it with with κ = 1 and O(1/ϵ4) queries. We give the first result for higher dimensions n. Perhaps surprisingly, our algorithm completely evades the “curse of dimensionality”: for any n and any κ> 4 π ≈ 1.27 we give a test that uses O(1/ϵ) queries. For small n we have improved bounds. For n = 1 we can achieve κ = 1 with O(1/ϵ3.5) queries (slightly improving [BBBY12]), or any κ> 1 with O(1/ϵ) queries (improving [KR98]). For n = 2, 3 we obtain κ ≈ 1.08, 1.125 respectively, with O(1/ϵ) queries. Getting an arbitrary κ> 1 for n> 1 remains an open problem.