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 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 problem, in which we are given a set S of n points in the pane, 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(np-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.
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