On piercing sets of axis-parallel rectangles and rings

Abstract

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, and is applied to find a better rectilinear 5-center algorithm. We improve the existing algorithm for general (but fixed) p, and we also extend our algorithms to higher dimensional space. We also consider the problem of piercing a set of rectangular rings.