On piercing sets of axis-parallel rectangles and rings

We consider the p-piercing problem for axis-parallel rectangles. We are given a collection of axis-parallel rectangles in the plane and wish to determine whether there exists a set of p points whose union intersects all the given rectangles. We present efficient algorithms for finding a piercing set (i.e., a set of p points as above) for values of p = 1,2,3,4,5. The result for 4 and 5-piercing improves an existing result of O(n log³ n) and O(n log⁴ n) to O(n log n) time. The result for 5-piercing can be applied find an O(n log² n) time algorithm for planar rectilinear 5-center problem, in which we are given a set S of n points in the plane, and wish to find 5 axis-parallel congruent squares of smallest possible size whose union covers S. We improve the existing algorithm for general (but fixed) p to O(n^p-4 log n) running time, and we also extend our algorithms to higher dimensional space. We also consider the problem of piercing a set of rectangular rings.