A simple linear algorithm for computing rectangle 3-centers

Rectangular p-centers of a nite planar point set P are the centers of at most p axis-parallel congruent squares of minimal size covering P . We give a simple linear time algorithm based on linear selection for the case p = 3. A linear algorithm for this problem is already known [7]. But it makes use of an LP -type [6] formulation of the problem with high combinatorial dimension (roughly 40) which makes it unlikely to perform well in an actual implementation. The motivation for our algorithm is such an implementation. 1 Theoretical Results Let P be a set of points in the plane and denote its bounding box by BP . The 3-radius of P is the minimal % 2 R such that P can be covered by three axis-parallel congruent squares of side length 2%. Excluding trivial cases we have BP = [xl; xr] [yb; yt]; xl < xr; yb < yt. For the sake of simplicity let us furthermore assume that P is in general position, i.e. no two points have a common x{ or y-coordinate nor the same jj jj1-distance to one of the corners of BP . We will rst repeat a number of simple observations as listed in [7].