On Rectilinear Partitions with Minimum Stabbing Number

Let S be a set of n points in R, and let r be a parameter with 1 6 r 6 n. A rectilinear r-partition for S is a collection Ψ(S) := {(S1, b1), . . . , (St, bt)}, such that the sets Si form a partition of S, each bi is the bounding box of Si, and n/2r 6 |Si| 6 2n/r for all 1 6 i 6 t. The (rectilinear) stabbing number of Ψ(S) is the maximum number of bounding boxes in Ψ(S) that are intersected by an axis-parallel hyperplane h. We study the problem of finding an optimal rectilinear r-partition—a rectilinear partition with minimum stabbing number—for a given set S. We obtain the following results. – Computing an optimal partition is np-hard already in R. – There are point sets such that any partition with disjoint bounding boxes has stabbing number Ω(r1−1/d), while the optimal partition has stabbing number 2. – An exact algorithm to compute optimal partitions, running in polynomial time if r is a constant, and a faster 2-approximation algorithm. – An experimental investigation of various heuristics for computing rectilinear r-partitions in R.