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Balanced Aspect Ratio Trees: Combining the Advantages of kd Trees and Octrees
"... Given a set S of n points in R^d, we show, for fixed d, how to construct in O(n log n) time a data structure we call the Balanced Aspect Ratio (BAR) tree. A BAR tree is a binary space partition tree on S that has O(logn) depth and in which every region is convex and “fat ” (that is, has a bounded as ..."
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Cited by 55 (8 self)
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Given a set S of n points in R^d, we show, for fixed d, how to construct in O(n log n) time a data structure we call the Balanced Aspect Ratio (BAR) tree. A BAR tree is a binary space partition tree on S that has O(logn) depth and in which every region is convex and “fat ” (that is, has a bounded aspect ratio). While previous hierarchical data structures, such as kd trees, quadtrees, octrees, fairsplit trees, and balanced box decompositions can guarantee some of these properties, we know of no previous data structure that combines alI of these properties simultaneously. The BAR tree data structure has numerous applications ranging from solving several geometric searching problems in fixed dimensional space to aiding in the visualization of graphs and threedimensional worlds.
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 29 (15 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...
Lower Bounds For Kinetic Planar Subdivisions
 IN PROC. 15TH ACM SYMP. ON COMPUTATIONAL GEOMETRY
, 1999
"... We revisit the notion of kinetic efficiency for noncanonicallydefined discrete attributes of moving data, like binary space partitions and triangulations. Under reasonable computational models, we obtain lower bounds on the minimum amount of work required to maintain any binary space partition o ..."
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Cited by 16 (9 self)
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We revisit the notion of kinetic efficiency for noncanonicallydefined discrete attributes of moving data, like binary space partitions and triangulations. Under reasonable computational models, we obtain lower bounds on the minimum amount of work required to maintain any binary space partition of moving segments in the plane or any Steiner triangulation of moving points in the plane. Such lower bounds the first to be obtained in the kinetic contextare necessary to evaluate the efficiency of kinetic data structures when the attribute to be maintained is not canonically defined.
Practical techniques for constructing binary space partitions for orthogonal rectangles
 In Proc. 13th Annu. ACM Sympos. Comput. Geom
, 1997
"... We present the rst systematic comparison of the performance of algorithms that construct Binary Space Partitions for orthogonal rectangles in R 3. We compare known algorithms with our implementation of a variant of a recent algorithm of Agarwal et al. [1]. We show via an empirical study that their a ..."
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Cited by 11 (0 self)
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We present the rst systematic comparison of the performance of algorithms that construct Binary Space Partitions for orthogonal rectangles in R 3. We compare known algorithms with our implementation of a variant of a recent algorithm of Agarwal et al. [1]. We show via an empirical study that their algorithm constructs BSPs of nearlinear size in practice and performs better than most of the other algorithms in the literature. 1
On the Exact Size of the Binary Space Partitioning of Sets of Isothetic Rectangles with Applications (Extended Abstract)
 SIAM Journal of Discrete Mathematics
, 2000
"... ) Abstract We show an upper bound of 3n on size of the Binary Space Partitioning (BSP) tree for a set of n isothetic rectangles, and an upper bound of 2n if the rectangles tile the underlying space. This improves the bound of 12n from [PY92] and 4n in [NW95, dAF92]. The BSP tree is one of the m ..."
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Cited by 9 (5 self)
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) Abstract We show an upper bound of 3n on size of the Binary Space Partitioning (BSP) tree for a set of n isothetic rectangles, and an upper bound of 2n if the rectangles tile the underlying space. This improves the bound of 12n from [PY92] and 4n in [NW95, dAF92]. The BSP tree is one of the most popular data structures and even "small" factor improvements of 4/3 or 2 we show improves the performance of applications relying on the BSP tree. Furthermore, our upper bounds yield improved approximation algorithms for several rectangular tiling problems in the literature. We also a show a lower bound of 2n in the worst case for a BSP for n isothetic rectangles, and a lower bound of 1.5n if they must form a tiling of the space. 1 Introduction Binary Space Partitioning (BSP) for a collection of geometric objects on two dimensional plane (we restrict ourselves to two dimensional space throughout the paper) is defined as follows. The space is divided into two (not necessarily equal...
On the union of κround objects in three and four dimensions
 Geom
, 2004
"... A compact set c in R d is κround if for every point p ∈ ∂c there exists a closed ball that contains p, is contained in c, and has radius κ diam c. We show that, for any fixed κ> 0, the combinatorial complexity of the union of n κround, not necessarily convex objects in R 3 (resp., in R 4) of const ..."
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Cited by 9 (6 self)
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A compact set c in R d is κround if for every point p ∈ ∂c there exists a closed ball that contains p, is contained in c, and has radius κ diam c. We show that, for any fixed κ> 0, the combinatorial complexity of the union of n κround, not necessarily convex objects in R 3 (resp., in R 4) of constant description complexity is O(n 2+ε) (resp., O(n 3+ε)) for any ε> 0, where the constant of proportionality depends on ε, κ, and the algebraic complexity of the objects. The bound is almost tight in the worst case. 1
Slice and Dice: A Simple, Improved Approximate Tiling Recipe
 In Proceedings of the 13th Annual ACMSIAM Symposium on Discrete Algorithms
, 2002
"... We are given a two dimensional array A[1 \Delta \Delta \Delta n; 1 \Delta \Delta \Delta n] where each A[i; j] stores a nonnegative number. A (rectangular) tiling of A is a collection of rectangular portions A[l \Delta \Delta \Delta r; t \Delta \Delta \Delta b], called tiles, such that no two tiles ..."
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Cited by 5 (3 self)
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We are given a two dimensional array A[1 \Delta \Delta \Delta n; 1 \Delta \Delta \Delta n] where each A[i; j] stores a nonnegative number. A (rectangular) tiling of A is a collection of rectangular portions A[l \Delta \Delta \Delta r; t \Delta \Delta \Delta b], called tiles, such that no two tiles overlap and each entry A[i; j] is contained in a tile. The weight of a tile is the sum of all array entries in it.
Efficient and Realistic Mobility and Channel Modeling for VANET Scenarios using OMNeT++ and INETFramework
, 2008
"... Mobility and channel modeling is a very crucial task for ..."
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Cited by 4 (0 self)
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Mobility and channel modeling is a very crucial task for
Linear BSP Trees for Sets of Hyperrectangles with Low Directional Density
 WSCG 2001 Conference Proceedings
, 2001
"... We consider the problem of constructing of binary space partitions #BSP# for a set S of n hyperrectangles in space with constant dimension. If the set S ful#lls the low directional density condition de#ned in this paper then the resultant BSP has O#n# size and it can be constructed in O#n log 2 ..."
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Cited by 3 (2 self)
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We consider the problem of constructing of binary space partitions #BSP# for a set S of n hyperrectangles in space with constant dimension. If the set S ful#lls the low directional density condition de#ned in this paper then the resultant BSP has O#n# size and it can be constructed in O#n log 2 n# time in R 3 . The low directional density condition de#nes a new class of objects which we are able to construct a linear BSP for. The method is quite simple and it should be appropriate for practical implementation. Keywords: BSP, partitioning, hyperrectangle 1
Binary space partitions of orthogonal subdivisions
 IN PROCEEDINGS OF THE 2004 ACM SYMPOSIUM ON COMPUTATIONAL GEOMETRY
, 2004
"... 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 ..."
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Cited by 3 (0 self)
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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