We consider the problem of constructing binary space partitions (BSPs) for orthogonal subdivisions (space filling packings of boxes) in d-space. We show that a subdivision with n boxes can be refined into a BSP of size O(n d+13), for all d * 3, and that such a partition can be computed in time O(K log n), where K is the size of the BSP produced. Our upper bound on the BSP size is tight for 3-dimensional subdivisions; in higher dimensions, this is the first nontrivial result for general full-dimensional boxes. We also present a lower bound construction for a subdivision of n boxes in d-space that requires a BSP of size \Omega (nfi(d)), where fi(d) converges to (1 + p 5)=2 as d! 1.

A binary space partition tree is a data structure for the representation of a set of objectsin space. It found an increasing number of applications over the last decades. In recent years, intensifying research focused on its combinatorial properties, which affect directly the efficiency of applications. Important advances were made on binary space partitions for disjoint line segments in the plane and for axis-aligned objects in higher dimensions. New research directions were also initiated on some realistic polygonal scenes and on kinetic binary space partitions. This paper attempts to give an overview of these results and reiterates some of the most pressing open problems.

Abstract: We present a novel sorting algorithm, Vis-Sort, to sort 1D and 3D geometric elements. Given a set of acyclic and non-intersecting 3D geometric primitives, Vis-Sort computes the visibility ordering from a viewpoint. The running time of our algorithm is dependent upon the degree of sortedness in the 3D sequence and is bounded by O(�Y �n), where n is the number of primitives and �Y � is the Knuth’s measure of disorder. The Knuth’s measure of disorder computes the minimum number of elements that need to be removed from the sequence for the remaining sequence to be sorted [35]. Vis-Sort exploits the spatial and temporal coherence between successive instances in a dynamic environment and performs incremental computations. Our algorithm requires no preprocessing and is applicable to all kind of models, including polygon soups and deformable models. We have used our algorithm for order-independent transparency computations in high-depth complexity environments and performing N-body collision culling in dynamic environments. We have implemented our algorithm and tested the system on a PC with a 3.4 GHz Pentium IV CPU with an NVIDIA GeForce FX 6800 Ultra GPU and applied it to complex environments with tens or hundreds of thousands of polygons. Our algorithm can compute a visibility ordering among the objects and triangles at interactive frame rates.

In this paper we consider the Rectilinear Minimum Link-Distance Problem in Three Dimensions. The prob-lem is well studied in two dimensions, but is relatively unexplored in higher dimensions. We solve the problem in O ( n log n) time, where n is the number of corners among all obstacles, and is the size of a BSP decomposition of the space containing the obstacles. It has been shown that in the worst case = (n3=2), giving us an overall worst case time of O(n5=2 log n). Previously known algorithms have had worst-case running times of (n³).

Abstract. In this paper we consider the Rectilinear Minimum Bends Path Problem among rectilinear obstacles in three dimensions. The problem is well studied in two dimensions, but is relatively unexplored in higher dimensions. We give an algorithm which solves the problem in worst-case O(βn log n) time, where n is the number of corners among all obstacles, and β is the size of a BSP decomposition of the space containing the obstacles. It has been shown that in the worst case β = Θ(n 3/2), giving us an overall worst case time of O(n 5/2 log n). However in many practical circumstances we will have β ≈ O(n). Previously known algorithms have a worst-case running time of O(n 3).

The binary space partition (for short, BSP) is a scheme for subdividing the ambient space Rd into open convex sets (called cells) by hyperplanes in a recursive fashion. Each subdivision step for a cell results in two cells, in which the process may continue, independently of other cells, until a stopping criterion is met. The binary recursion tree, also called BSP-tree, is traditionally used as a data structure in computer graphics for efficient rendering of polyhedral scenes. Each node v of the BSP-tree, except for the leaves, corresponds to a cell Cv ⊆ Rd and a partitioning hyperplane Hv. The cell of the root r is Cr = Rd, and the two children of a node v correspond to Cv∩H−v and Cv∩H+v, where H−v and H v denote the open halfspaces bounded by Hv. Refer to Fig. 1. A binary space partition for a set of n pairwise disjoint (typically polyhedral) objects in Rd is a BSP where the space is recursively partitioned until each cell intersects at most one object. When the BSP-tree is used as a data structure, every leaf v stores the fragment of at most one object clipped in the cell Cv, and every interior node v stores the fragments of any lower-dimensional objects that lie in Cv ∩Hv. 2aa b c d b1 b2 c1 c2 d