Computing the minimum Hausdorff distance for point sets under translation

We consider the problem of computing a translation that minimizes the Hausdorff distance between two sets of points. For points in @@@@1 in the worst case there are ⊖(mn) translations at which the Hausdorff distance is a local minimum, where m is the number of points in one set and n is the number in the other. For points in @@@@2 there are ⊖(mn(m + n)) such local minima. We show how to compute the minimal Hausdorff distance in time Ο(mn log mn) for points in @@@@1 and in time Ο(m2n2α(mn)) for points in @@@@2. The results for the one-dimensional case are applied to the problem of comparing polygons under general affine transformations, where we extend the recent results of Arkin et al on polygon resemblance under rigid body motion. The two-dimensional case is closely related to the problem of finding an approximate congruence between two points sets under translation in the plane, as considered by Alt et al.