Computing the smallest k-enclosing circle and related problems

We present an efficient algorithm for solving the "smallest k-enclosing circle" (kSC) problem: Given a set of n points in the plane and an integer k ≤ n, find the smallest disk containing k of the points. We present two solutions. When using O(nk) storage, the problem can be solved in time O(nk log² n). When only O(n log n) storage is allowed, the running time is O(nk log² n log n/k). We also extend our technique to obtain efficient solutions of several related problems (with similar time and storage bounds). These related problems include: finding the smallest homothetic copy of a given convex polygon P which contains k points from a given planar set, and finding the smallest disk intersecting k segments from a given planar set of non-intersecting segments.