On rectilinear partitions with minimum stabbing number

Abstract

Let S be a set of n points in ℝd, and let r be a parameter with 1 ≤ r ≤ 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 ≤ |Si| ≤ 2n/r for all 1 ≤ i ≤ 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 ℝ2. - 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 ℝ2. © 2011 Springer-Verlag.