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 k-d trees, quadtrees, octrees, fair-split 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 three-dimensional worlds.

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 xy-plane. 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...

We revisit the notion of kinetic efficiency for non-canonically-defined 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 context---are necessary to evaluate the efficiency of kinetic data structures when the attribute to be maintained is not canonically defined.

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 near-linear size in practice and performs better than most of the other algorithms in the literature. 1

) 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...

A compact set c in R d is κ-round 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 κ-round, 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

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 non-negative 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.

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

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

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.