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 $k \in \omega(n / \log n)$.

翻译：我们考虑在具有$m$个顶点（其中$r$个为反射顶点）的简单多边形内，从$n$个点中寻找包含至少$k$个点的最小半径测地圆盘问题，并将此类圆盘称为SKEG圆盘。我们提出一种基于高阶测地Voronoi图计算SKEG圆盘的算法，忽略多对数因子后其最坏时间复杂度为$O(k^{2} n + k^{2} r + \min(kr, r(n-k)) + m)$。随后我们提出两种$2$-近似算法，可找到包含至少$k$个点且半径不超过SKEG圆盘两倍的圆盘。第一种算法以高概率在$O((n^{2} / k) \log n \log r + m)$最坏时间复杂度和$O(n + m)$空间复杂度内计算$2$-近似解。第二种算法在$O(n \log^{2} n \log r + m)$期望时间复杂度和$O(n + m)$期望空间复杂度内运行，且时间复杂度与$k$无关。值得注意的是，当$k \in \omega(n / \log n)$时，第一种算法速度更快。