Results 1  10
of
33
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 \Sigma(x ..."
Abstract

Cited by 97 (15 self)
 Add to MetaCart
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)
"... ..."
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 ..."
Abstract

Cited by 55 (8 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 32 (8 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 29 (15 self)
 Add to MetaCart
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...
Binary space partitions for fat rectangles
 in: Proc. 37th Annu. IEEE Sympos. Found. Comput. Sci
, 1996
"... sizen2O(plogn) We consider the practical problem of constructing binary space partitions (BSPs) for a setSofnorthogonal, nonintersecting, twodimensional rectangles inIR3such that the aspect ratio of each rectangle inSis at most, for some constant1. We present ann2O(plogn)time algorithm to build a ..."
Abstract

Cited by 23 (6 self)
 Add to MetaCart
sizen2O(plogn) We consider the practical problem of constructing binary space partitions (BSPs) for a setSofnorthogonal, nonintersecting, twodimensional rectangles inIR3such that the aspect ratio of each rectangle inSis at most, for some constant1. We present ann2O(plogn)time algorithm to build a binary space partition of forS. We also show that ifmof thenrectangles inShave aspect ratios greater than, we can construct a BSP of sizenpm2O(plogn)forSinnpm2O(plogn)time. The constants of proportionality in the bigoh terms are linear inlog. We extend these results to cases in which the input contains nonorthogonal or intersecting objects.
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 ..."
Abstract

Cited by 20 (10 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 19 (6 self)
 Add to MetaCart
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 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 axispar ..."
Abstract

Cited by 18 (0 self)
 Add to MetaCart
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.