On piercing sets of axis-parallel rectangles and rings

We consider the p-piercing problem for axis-parallel rectangles. We are giv en a collection of axis-parallel rectangles in the plane and wish to determ ine whether there exists a set of p points whose union intersects all the g iven 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 re sult for 4 and 5-piercing improves an existing result of O(n log(3) n) and O(n log(4) n) to O(n log n) time. The result for 5-piercing can be applied find an O(n log(2) n) time algorithm for planar rectilinear 5-center proble m, 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 co vers 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 dimen sional space. We also consider the problem of piercing a set of rectangular rings.