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.
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Computing an optimal partition is np-hard already in \({\Bbb R}^2\).
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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.
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An exact algorithm to compute optimal partitions, running in polynomial time if r is a constant, and a faster 2-approximation algorithm.
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An experimental investigation of various heuristics for computing rectilinear r-partitions in \({\Bbb R}^2\).
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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.
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de Berg, M., Khosravi, A., Verdonschot, S., van der Weele, V. (2011). On Rectilinear Partitions with Minimum Stabbing Number. In: Dehne, F., Iacono, J., Sack, JR. (eds) Algorithms and Data Structures. WADS 2011. Lecture Notes in Computer Science, vol 6844. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22300-6_26
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DOI: https://doi.org/10.1007/978-3-642-22300-6_26
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