Exact Size of Binary Space Partitionings and Improved Rectangle Tiling Algorithms

Abstract

We prove the following upper and lower bounds on the exact size of binary space partition (BSP) trees for a set of n isothetic rectangles in the plane: An upper bound of 3n-1 in general, and an upper bound of 2n-1 if the rectangles tile the underlying space. This improves the upper bounds of 4n in [V. Hai Nguyen and P. Widmayer, Binary Space Partitions for Sets of Hyperrectangles, Lecture Notes in Comput. Sci. 1023, Springer-Verlag, Berlin, 1995; F. d'Amore and P. G. Franciosa, Inform. Process. Lett., 44 (1992), pp. 255--259]. A BSP satisfying the upper bounds can be constructed in O(n log n) time. A worst-case lower bound of 2n-o(n) in general, and $\frac{3n}{2}-o(n)$ if the rectangles form a tiling. The BSP tree is one of the most popular data structures in computational geometry, and hence even small factor improvements of $\frac{4}{3}$ or 2 on the previously known upper bounds that we show improve the performances of applications relying on the BSP tree. As an illustration, we present improved approximation algorithms for certain dual rectangle tiling problems using our upper bounds on the size of the BSP trees.