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55
Efficient Binary Space Partitions for HiddenSurface Removal and Solid Modeling
, 1990
"... We consider schemes for recursively dividing a set of geometric objects by hyperplanes until all objects are separated. Such a binary space partition, or BSP, is naturally considered as a binary tree where each internal node corresponds to a division. The goal is to choose the hyperplanes properly ..."
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Cited by 102 (0 self)
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We consider schemes for recursively dividing a set of geometric objects by hyperplanes until all objects are separated. Such a binary space partition, or BSP, is naturally considered as a binary tree where each internal node corresponds to a division. The goal is to choose the hyperplanes properly so that the size of the BSP, i.e., the number of resulting fragments of the objects, is minimized. For the twodimensional case, we construct BSPs of size O(n log n) for n edges, while in three dimensions, we obtain BSPs of size O(n²) for n planar facets and prove a matching lower bound of Ω(n²). Two applications of efficient BSPs are given. The first is an O(n²)sized data structure for implementing a hiddensurface removal scheme of Fuchs et al. [6]. The second application is in solid modeling: given a polyhedron described by its n faces, we show how to generate an O(n²)sized CSG (constructivesolidgeometry) formula whose literals correspond to halfspaces supporting the faces of the polyhedron. The best previous results for both of these problems were O(n³).
Surface Approximation and Geometric Partitions
 IN PROC. 5TH ACMSIAM SYMPOS. DISCRETE ALGORITHMS
, 1994
"... Motivated by applications in computer graphics, visualization, and scientific computation, we study the computational complexity of the following problem: Given a set S of n points sampled from a bivariate function f(x; y) and an input parameter " ? 0, compute a piecewise linear function \Si ..."
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Cited by 102 (15 self)
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Motivated by applications in computer graphics, visualization, and scientific computation, we study the computational complexity of the following problem: Given a set S of n points sampled from a bivariate function f(x; y) and an input parameter " ? 0, compute a piecewise linear function \Sigma(x; y) of minimum complexity (that is, a xymonotone polyhedral surface, with a minimum number of vertices, edges, or faces) such that j\Sigma(x p ; y p ) \Gamma z p j "; for all (x p ; y p ; z p ) 2 S: We prove that the decision version of this problem is NPHard . The main result of our paper is a polynomialtime approximation algorithm that computes a piecewise linear surface of size O(K o log K o ), where K o is the complexity of an optimal surface satisfying the constraints of the problem. The technique
Approximation Algorithms for Geometric Tour and Network Design Problems (Extended Abstract)
"... ..."
Linear Size Binary Space Partitions for Uncluttered Scenes
 Algorithmica
, 1998
"... We describe a new and simple method for constructing binary space partitions in arbitrary dimensions. We also introduce the concept of uncluttered scenes, which are scenes with a certain property that we suspect many realistic scenes exhibit, and we show that our method constructs a BSP of size O ..."
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Cited by 33 (9 self)
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We describe a new and simple method for constructing binary space partitions in arbitrary dimensions. We also introduce the concept of uncluttered scenes, which are scenes with a certain property that we suspect many realistic scenes exhibit, and we show that our method constructs a BSP of size O(n) for an uncluttered scene consisting of n objects. The construction time is O(n log n). Because any set of disjoint fat objects is uncluttered, our result implies an efficient method to construct a linear size BSP for fat objects. We use our BSP to develop a data structure for point location in uncluttered scenes. The query time of our structure is O(log n), and the amount of storage is O(n). This result can in turn be used to perform range queries with nottoosmall ranges in scenes consisting of disjoint fat objects or, more generally, in socalled lowdensity scenes. 1 Introduction Many geometric problems can be solved more easily if a decomposition of the space of interest in...
Cylindrical Static and Kinetic Binary Space Partitions
, 1997
"... We describe the first known algorithm for efficiently maintaining a Binary Space Partition (BSP) for n continuously moving segments in the plane. Under reasonable assumptions on the motion, we show that the total number of times the BSP changes is O(n 2 ), and that we can update the BSP in O(log n ..."
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Cited by 30 (16 self)
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We describe the first known algorithm for efficiently maintaining a Binary Space Partition (BSP) for n continuously moving segments in the plane. Under reasonable assumptions on the motion, we show that the total number of times the BSP changes is O(n 2 ), and that we can update the BSP in O(log n) expected time per change. We also consider the problem of constructing a BSP for n triangles in R 3 . We present a randomized algorithm that constructs a BSP of expected size O(n 2 ) in O(n 2 log 2 n) expected time. We also describe a deterministic algorithm that constructs a BSP of size O((n + k) log n) and height O(log n) in O((n + k) log 2 n) time, where k is the number of intersection points between the edges of the projections of the triangles onto the xyplane. 1 Introduction The Binary Space Partition (BSP, also known as BSP tree), originally proposed by Schumacker et al. [26] and further refined by Fuchs et al. [16], is a hierarchical partitioning of space widely used i...
Kinetic Binary Space Partitions for Intersecting Segments and Disjoint Triangles (Extended Abstract)
, 1998
"... We describe randomized algorithms for efficiently maintaining a binary space partition of continuously moving, possibly intersecting, line segments in the plane, and of continuously moving but disjoint triangles in space. Our twodimensional BSP has depth O(log n) and size O(n log n + k) and can be ..."
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Cited by 22 (10 self)
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We describe randomized algorithms for efficiently maintaining a binary space partition of continuously moving, possibly intersecting, line segments in the plane, and of continuously moving but disjoint triangles in space. Our twodimensional BSP has depth O(log n) and size O(n log n + k) and can be constructed in expected O(n log² n + k log n) time, where k is the number of intersecting pairs. We can detect combinatorial changes to our BSP caused by the motion of the segments, and we can update our BSP in expected O(log n) time per change. Our threedimensional BSP has depth O(log n), size O(n log² n+k 0 ), construction time O(n log³ n+k 0 log n), and update time O(log² n) (all expected), where k 0 is the number of intersections between pairs of edges in the xy projection of the triangles. Under reasonable assumptions about the motion of the segments or triangles, the expected number of number o...
New Results on Binary Space Partitions in the Plane
 COMPUT. GEOM. THEORY APPL
, 1994
"... We prove the existence of linear size binary space partitions for sets of objects in the plane under certain conditions that are often satisfied in practical situations. In particular, we construct linear size binary space partitions for sets of fat objects, for sets of line segments where the ra ..."
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Cited by 20 (7 self)
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We prove the existence of linear size binary space partitions for sets of objects in the plane under certain conditions that are often satisfied in practical situations. In particular, we construct linear size binary space partitions for sets of fat objects, for sets of line segments where the ratio between the lengths of the longest and shortest segment is bounded by a constant, and for homothetic objects. For all cases we also show how to turn the existence proofs into efficient algorithms.
Binary space partitions for fat rectangles
 IN: PROC. 37TH ANNU. IEEE SYMPOS. FOUND. COMPUT. SCI
, 1996
"... We consider the practical problem of constructing binary space partitions (BSPs) for a set S of n orthogonal, nonintersecting, twodimensional rectangles inIR3such that the aspect ratio of each rectangle inSis at most, for some constant 1. We present an n2^O(p log n)time algorithm to build a bina ..."
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Cited by 20 (6 self)
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We consider the practical problem of constructing binary space partitions (BSPs) for a set S of n orthogonal, nonintersecting, twodimensional rectangles inIR3such that the aspect ratio of each rectangle inSis at most, for some constant 1. We present an n2^O(p log n)time algorithm to build a binary space partition of for S. We also show that if m of the n rectangles in S have aspect ratios greater than, we can construct a BSP of size npm2O(plogn) for S in npm2O(plogn) time. The constants of proportionality in the bigoh terms are linear in log. We extend these results to cases in which the input contains nonorthogonal or intersecting objects.
Binary Space Partitions for AxisParallel Segments, Rectangles, and Hyperrectangles
 IN PROC. 17TH ANNU. ACM SYMPOS. COMPUT. GEOM
, 2001
"... We provide a variety of new results, including upper and lower bounds, as well as simpler proof techniques for the ecient construction of binary space partitions (BSP's) of axisparallel segments, rectangles, and hyperrectangles. (a) A consequence of the analysis in [1] is that any set of n axi ..."
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Cited by 19 (1 self)
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We provide a variety of new results, including upper and lower bounds, as well as simpler proof techniques for the ecient construction of binary space partitions (BSP's) of axisparallel segments, rectangles, and hyperrectangles. (a) A consequence of the analysis in [1] is that any set of n axisparallel and pairwisedisjoint line segments in the plane admits a binary space partition of size at most 2n 1. We establish a worstcase lower bound of 2n o(n) for the size of such a BSP, thus showing that this bound is almost tight in the worst case. (b) We give an improved worstcase lower bound of 9 4 n o(n) on the size of a BSP for isothetic pairwise disjoint rectangles. (c) We present simple methods, with equally simple analysis, for constructing BSP's for axisparallel segments in higher dimensions, simplifying the technique of [9] and improving the constants. (d) We obtain an alternative construction (to that in [9]) of BSP's for collections of axisparallel rectangles in 3space. (e) We present a construction of BSP's of size O(n 5=3 ) for n axisparallel pairwise disjoint 2rectangles in R 4 , and give a matching worstcase lower bound of n 5=3 ) for the size of such a BSP. (f) We extend the results of [9] to axisparallel kdimensional rectangles in R d , for k < d=2, and obtain a worstcase tight bound of (n d=(d k) ) for the size of a BSP of n rectangles. Both upper and lower bounds also hold for d=2 k d 1 if we allow the rectangles to intersect.