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21
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...
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 14 (8 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.
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 12 (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...
Approximate Range Searching Using Binary Space Partitions
"... We show how any BSP tree TP for the endpoints of a set of n disjoint segments in the plane can be used to obtain a BSP tree of size O(n · depth(TP)) for the segments themselves, such that the rangesearching efficiency remains almost the same. We apply this technique to obtain a BSP tree of size O(n ..."
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Cited by 11 (3 self)
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We show how any BSP tree TP for the endpoints of a set of n disjoint segments in the plane can be used to obtain a BSP tree of size O(n · depth(TP)) for the segments themselves, such that the rangesearching efficiency remains almost the same. We apply this technique to obtain a BSP tree of size O(n log n) such that εapproximate range searching queries with any constantcomplexity convex query range can be answered in O(minε>0{(1/ε) + kε} log n) time, where kε is the number of segments intersecting the εextended range. The same result can be obtained for disjoint constantcomplexity curves, if we allow the BSP to use splitting curves along the given curves. We also describe how to construct a linearsize BSP tree for lowdensity scenes consisting of n objects in R d such that εapproximate range searching with any constantcomplexity convex query range can be done in O(log n + minε>0{(1/ε d−1) + kε}) time. Finally we show how to adapt our structures so that they become I/Oefficient.
State of the Union (of Geometric Objects)
 CONTEMPORARY MATHEMATICS
"... Let C be a set of geometric objects in R d. The combinatorial complexity of the union of C is the total number of faces of all dimensions on its boundary. We survey the known upper bounds on the complexity of the union of n geometric objects satisfying various natural conditions. These bounds play ..."
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Cited by 11 (7 self)
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Let C be a set of geometric objects in R d. The combinatorial complexity of the union of C is the total number of faces of all dimensions on its boundary. We survey the known upper bounds on the complexity of the union of n geometric objects satisfying various natural conditions. These bounds play a central role in the analysis of many geometric algorithms, and the techniques used to attain these bounds are interesting in their own right.
Practical techniques for constructing binary space partitions for orthogonal rectangles
 IN PROC. 13TH ANNU. ACM SYMPOS. COMPUT. GEOM
, 1997
"... We present the first systematic comparison of the performance of algorithms that construct Binary Space Partitions for orthogonal rectangles in R³. 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 ..."
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Cited by 10 (0 self)
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We present the first systematic comparison of the performance of algorithms that construct Binary Space Partitions for orthogonal rectangles in R³. 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.
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 co ..."
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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
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 8 (0 self)
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Mobility and channel modeling is a very crucial task for
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 6 (2 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.
State of the Union (of Geometric Objects): A Review
, 2007
"... Let C be a set of geometric objects in R d. The combinatorial complexity of the union U(C) of C is the total number of faces of all dimensions, of the arrangement of the boundaries of the objects, which lie on its boundary. We survey the known upper bounds on the complexity of the union of n geometr ..."
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Cited by 6 (1 self)
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Let C be a set of geometric objects in R d. The combinatorial complexity of the union U(C) of C is the total number of faces of all dimensions, of the arrangement of the boundaries of the objects, which lie on its boundary. We survey the known upper bounds on the complexity of the union of n geometric objects satisfying various natural conditions. These problems play a central role in the design and analysis of many geometric algorithms arising in robotics, molecular modeling, solid modeling, and shape matching, and the techniques used for their solutions are interesting in their own right.