On Rectilinear Partitions with Minimum Stabbing Number

Abstract

Let S be a set of n points in ${\Bbb R}^d$, and let r be a parameter with 1 ≤ r ≤ n. A rectilinear r-partition for S is a collection Ψ(S) : = {(S ₁,b ₁),…,(S _t ,b _t )}, such that the sets S _i form a partition of S, each b _i is the bounding box of S _i , and n/2r ≤ |S _i | ≤ 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 ${\Bbb R}^2$.

• There are point sets such that any partition with disjoint bounding boxes has stabbing number Ω(r ^{1 − 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 ${\Bbb R}^2$.

The research by Amirali Khosravi was supported by the Netherlands’ Organisation for Scientific Research (NWO) under project no. 639.023.301 and project no. 612.000.631.