Approximation Algorithms for Minimum-Width Annuli and Shells

Let S be a set of n points in ℝ^d. The "roundness" of S can be measured by computing the width ωz.ast; = ω*(S) of the thinnest spherical shell (or annulus in ℝ²) that contains S. This paper contains two main results related to computing an approximation of ω*: (i) For d = 2, we can compute in O(n log n) time an annulus containing S whose width is at most 2ω*(S). We extend this algorithm, so that, for any given parameter ε > 0, an annulus containing S whose width is at most (1 + ε)ω* is computed in time O(n log n + n/ε²). (ii) For d ≥ 3, given a parameter ε > 0, we can compute a shell containing S of width at most (1 + ε)ω* either in time O((n/ε^d) log(Δ/ω*ε)) or in time O((n/ε^d-2)(log n + 1/ε) log(Δ/ω*ε)), where Δ is the diameter of S.