Optimal Binary Space Partitions in the Plane
An optimal bsp for a set S of disjoint line segments in the plane is a bsp for S that produces the minimum number of cuts. We study optimal bsps for three classes of bsps, which differ in the splitting lines that can be used when partitioning a set of fragments in the recursive partitioning process: free bsps can use any splitting line, restricted bsps can only use splitting lines through pairs of fragment endpoints, and auto-partitions can only use splitting lines containing a fragment. We obtain the two following results:
It is np-hard to decide whether a given set of segments admits an auto-partition that does not make any cuts.
An optimal restricted bsp makes at most 2 times as many cuts as an optimal free bsp for the same set of segments.
KeywordsFull Version Splitting Line Negative Clause Variable Gadget Planar Instance
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- 3.de Berg, M., Mumford, E., Speckmann, B.: Optimal BSPs and rectilinear cartograms. In: Proc. 14th Int. Symp. Adv. Geographic Inf. Syst. (ACM-GIS), pp. 19–26 (2006)Google Scholar
- 4.Clairbois, X.: On Optimal Binary Space Partitions. MSc thesis, TU Eindhoven (2006)Google Scholar
- 8.Tóth, C.D.: Binary space partitions: recent developments. In: Goodman, J.E., Pach, J., Welzl, E. (eds.) Combinatorial and Computational Geometry. MSRI Publications, vol. 52, pp. 525–552. Cambridge University Press, Cambridge (2005)Google Scholar
- 9.Katz, B., Rutter, I., Woeginger, G.: An algorithmic study of switch graphs. In: Paul, C. (ed.) WG 2009. LNCS, vol. 5911, pp. 226–237. Springer, Heidelberg (2009)Google Scholar