Approximating the Smallest k-Enclosing Geodesic Disc in a Simple Polygon

We consider the problem of finding a geodesic disc of smallest radius containing at least k points from a set of n points in a simple polygon that has m vertices, r of which are reflex vertices. We refer to such a disc as a SKEG disc. We present an algorithm to compute a SKEG disc using higher-order geodesic Voronoi diagrams with worst-case time O(k^2 n + k^2 r + min(kr, r(n-k)) + m) ignoring polylogarithmic factors. We then present two 2-approximation algorithms that find a geodesic disc containing at least k points whose radius is at most twice that of a SKEG disc. The first algorithm computes a 2-approximation with high probability in O((n^2 / k) log n log r + m) worst-case time with O(n + m) space. The second algorithm runs in O(n log^2 n log r + m) expected time using O(n + m) expected space, independent of k. Note that the first algorithm is faster when k ∈ω(n / log n).