Binary Space Partition for Orthogonal Fat Rectangles

Abstract We generate a binary space partition (BSP) of size O(n log8 n) and depth O(log4 n) for n orthogonal fat rectangles in three-space, improving earlier bounds of Agarwal et al. We also give a lower bound construction showing that the size of an orthogonal BSP for these objects is \Omega (n log n) in the worst case. 1 Introduction The binary space partition (BSP) is a data structure invented by the computer graphics community [9, 6]. It was used for fast rendering polygonal scenes and for shadow generation. Ever since it found many application in computer graphics, robotics, and computational and combinatorial geometry. A BSP is a recursive cutting scheme for a set of disjoint (or non-overlapping) polygonal scenes in the Euclidean space (R3). We split the bounding box of the polygons along a plane into two parts and then we partition recursively the two subproblems corresponding to the two subcells as long as the interior of a subcell intersects an input polygon. This partitioning procedure can be represented by a binary tree (called BSP tree) where every intermediate node stores a splitting plane and every leaf stores a convex cell. Similarly, the BSP can be defined for any set of (d \Gamma 1)-dimensional objects in Rd, d 2 N