Mapping Multiple Regions to the Grid with Bounded Hausdorff Distance.

We study a problem motivated by digital geometry: given a set of disjoint geometric regions, assign each region \(R_i\) a set of grid cells \(P_i\), so that \(P_i\) is connected, similar to \(R_i\), and does not touch any grid cell assigned to another region. Similarity is measured using the Hausdorff distance. We analyze the achievable Hausdorff distance in terms of the number of input regions, and prove asymptotically tight bounds for several classes of input regions.