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Binary Space Partitions of Orthogonal Subdivisions
"... We consider the problem of constructing binary space partitions (BSPs) for orthogonal subdivisions (space filling packings of boxes) in dspace. 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 ..."
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We consider the problem of constructing binary space partitions (BSPs) for orthogonal subdivisions (space filling packings of boxes) in dspace. 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 3dimensional subdivisions; in higher dimensions, this is the first nontrivial result for general fulldimensional boxes. We also present a lower bound construction for a subdivision of n boxes in dspace that requires a BSP of size \Omega (nfi(d)), where fi(d) converges to (1 + p 5)=2 as d! 1.
Binary Space Partitions  Recent Developments
, 2004
"... 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 applica ..."
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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 axisaligned 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.
Vissort: Fast visibility ordering of 3d geometric primitives
, 2004
"... Abstract: We present a novel sorting algorithm, VisSort, to sort 1D and 3D geometric elements. Given a set of acyclic and nonintersecting 3D geometric primitives, VisSort computes the visibility ordering from a viewpoint. The running time of our algorithm is dependent upon the degree of sortednes ..."
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Abstract: We present a novel sorting algorithm, VisSort, to sort 1D and 3D geometric elements. Given a set of acyclic and nonintersecting 3D geometric primitives, VisSort 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]. VisSort 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 orderindependent transparency computations in highdepth complexity environments and performing Nbody 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.
An O(n^5/2 log n) . . . the Rectilinear Minimum LinkDistance Problem . . .
"... In this paper we consider the Rectilinear Minimum LinkDistance Problem in Three Dimensions. The problem 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 ..."
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In this paper we consider the Rectilinear Minimum LinkDistance Problem in Three Dimensions. The problem 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 worstcase running times of (n³).
The Rectilinear Minimum Bends Path Problem in Three Dimensions
"... 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 worstcase O(βn log n) ti ..."
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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 worstcase 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 worstcase running time of O(n 3).
SumOriWork: No Years and Authors of Summarized Original Work Given Binary Space Partitions
"... 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 sto ..."
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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 BSPtree, is traditionally used as a data structure in computer graphics for efficient rendering of polyhedral scenes. Each node v of the BSPtree, 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 BSPtree 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 lowerdimensional objects that lie in Cv ∩Hv. 2aa b c d b1 b2 c1 c2 d