-The quadtree representation encodes a 2" by 2" binary image as a set of maximal blocks of l's or O's whose sizes and positions are powers of 2. With the aid of the quadtree, a hierarchy of approximations to the image can be defined. Several ways of doing this are described. The accuracy of these approximations is empirically evaluated by studying how fast estimates of the first few moments of ...

We analyse the distribution of the root pattern of randomly grown multidimensional point quadtrees. In particular, exact, recursive and asymptotic formulas are given for the expected arity of the root.

A central problem in the field of geometric algorithms is to compare algorithms that process vector-type data specified in the continuous plane with algorithms that process raster-type data specified in the digitized plane. A set of criteria is proposed that a digitization of a collection of line segments should satisfy in order to be said to represent the same structure as their continuous pla...

We propose a simple, direct approach for computing the expected cost of random partial match queries in random quadtrees. The approach gives not only an explicit expression for the leading constant in the asymptotic approximation of the expected cost but also more terms in the asymptotic expansion if desired.

The spatial join operation is benchmarked using variants of well-known spatial data structures such as the R-tree, R -tree, R-tree, and the PMR quadtree. The focus is on a spatial join with spatial output because the result of the spatial join frequently serves as input to subsequent spatial operations (i.e., a cascaded spatial join as would be common in a spatial spreadsheet). Thus, in additio...

The size of five trie-based methods of sorting large collections of line segments in a spatial database is investigated analytically using a random lines image model and geometric probability techniques. The methods are based on sorting the line segments with respect to the space that they occupy. Since the space is two-dimensional, the trie is formed by interleaving the bits corresponding to t...

The data presented by Light Detection and Ranging (LiDAR) systems are in a dense and accurate three dimensional pattern without point classification, such as trees, roads, and buildings. Extraction of boundary points is essential for recognizing buildings. However, it is complicated to process the LiDAR data due to its irregularity and a large number of collected data points. In order to find b...

Ralph Neininger University of Freiburg November 22, 1999 [summary by Elchanan Mossel] 1. Basic Algorithms We consider records which belong to a k dimensional region D = D1 Dk Rk. A le is a nite subset F of D. Given a query q 2 (D1 [ f g) (Dk [ f g), the objective is to nd all the records r 2 F such that ri = qi when qi 6= . The probabilistic assumption is that all the coordinates of the records...

We introduce dynamic smooth (a.k.a. balanced) compressed quadtrees with worst-case constant time updates in constant dimensions. We distinguish two versions of the problem. First, we show that quadtrees as a space-division data structure can be made smooth and dynamic subject to split and merge operations on the quadtree cells. Second, we show that quadtrees used to store a set of points in R c...