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Singularity Analysis, Hadamard Products, and Tree Recurrences
, 2003
"... We present a toolbox for extracting asymptotic information on the coecients of combinatorial generating functions. This toolbox notably includes a treatment of the eect of Hadamard products on singularities in the context of the complex Tauberian technique known as singularity analysis. As a consequ ..."
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Cited by 38 (10 self)
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We present a toolbox for extracting asymptotic information on the coecients of combinatorial generating functions. This toolbox notably includes a treatment of the eect of Hadamard products on singularities in the context of the complex Tauberian technique known as singularity analysis. As a consequence, it becomes possible to unify the analysis of a number of divideandconquer algorithms, or equivalently random tree models, including several classical methods for sorting, searching, and dynamically managing equivalence relations.
Hypergeometrics and the Cost Structure of Quadtrees
, 1995
"... Several characteristic parameters of randomly grown quadtrees of any dimension are analyzed. Additive parameters have expectations whose generating functions are expressible in terms of generalized hypergeometric functions. A complex asymptotic process based on singularity analysis and integral repr ..."
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Cited by 27 (3 self)
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Several characteristic parameters of randomly grown quadtrees of any dimension are analyzed. Additive parameters have expectations whose generating functions are expressible in terms of generalized hypergeometric functions. A complex asymptotic process based on singularity analysis and integral representations akin to Mellin transforms leads to explicit values for various structure constants related to path length, retrieval costs, and storage occupation.
Search costs in quadtrees and singularity perturbation asymptotics
 Discrete Comput. Geom
, 1994
"... Abstract. Quadtrees constitute a classical data structure for storing and accessing collections of points in multidimensional space. It is proved that, in any dimension, the cost of a random search in a randomly grown quadtree has logarithmic mean and variance and is asymptotically distributed as a ..."
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Cited by 21 (5 self)
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Abstract. Quadtrees constitute a classical data structure for storing and accessing collections of points in multidimensional space. It is proved that, in any dimension, the cost of a random search in a randomly grown quadtree has logarithmic mean and variance and is asymptotically distributed as a normal variable. The limit distribution property extends to quadtrees of all dimensions a result only known so far to hold for binary search trees. The analysis is based on a technique of singularity perturbation that appears to be of some generality. For quadtrees, this technique is applied to linear differential equations satisfied by intervening bivariate generating functions 1.
A Multivariate View of Random Bucket Digital Search Trees
, 2002
"... We take a multivariate view of digital search trees by studying the number of nodes of di#erent types that may coexist in a bucket digital search tree as it grows under an arbitrary memory management system. We obtain the mean of each type of node, as well as the entire covariance matrix between ..."
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Cited by 12 (6 self)
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We take a multivariate view of digital search trees by studying the number of nodes of di#erent types that may coexist in a bucket digital search tree as it grows under an arbitrary memory management system. We obtain the mean of each type of node, as well as the entire covariance matrix between types, whereupon weak laws of large numbers follow from the orders of magnitude (the norming constants include oscillating functions). The result can be easily interpreted for practical systems like paging, heaps and UNIX's buddy system. The covariance results call for developing a Mellin convolution method, where convoluted numerical sequences are handled by convolutions of their Mellin transforms. Furthermore, we use a method of moments to show that the distribution is asymptotically normal. The method of proof is of some generality and is applicable to other parameters like path length and size in random tries and Patricia tries.
On the Number of Descendants and Ascendants in Random Search Trees
, 1997
"... We consider here the probabilistic analysis of the number of descendants and the number of ascendants of a given internal node in a random search tree. The performance of several important algorithms on search trees is closely related to these quantities. For instance, the cost of a successful searc ..."
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Cited by 7 (3 self)
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We consider here the probabilistic analysis of the number of descendants and the number of ascendants of a given internal node in a random search tree. The performance of several important algorithms on search trees is closely related to these quantities. For instance, the cost of a successful search is proportional to the number of ascendants of the sought element. On the other hand, the probabilistic behavior of the number of descendants is relevant for the analysis of paged data structures and for the analysis of the performance of quicksort, when recursive calls are not made on small subfiles. We also consider the number of ascendants and descendants of a random node in a random search tree, i.e., the grand averages of the quantities mentioned above. We address these questions for standard binary search trees and for locally balanced search trees. These search trees were introduced by Poblete and Munro and are binary search trees such that each subtree of size 3 is balanced; in oth...
A Probabilistic Analysis of TrieBased Sorting of Large Collections of Line Segments in Spatial Databases
, 2000
"... The size of five triebased 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. ..."
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Cited by 6 (6 self)
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The size of five triebased 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 twodimensional, the trie is formed by interleaving the bits corresponding to the binary representation of the x and y coordinates of the underlying space and then testing two bits at each iteration. The result of this formulation yields a class of representations that are referred to as quadtrie variants, although they have been traditionally referred to as quadtree variants. The analysis differs from prior work in that it uses a detailed explicit model of the image instead of relying on modeling the branching process represented by the tree and leaving the underlying image unspecified. The analysis provides analytic expressions and bounds on the expected size of these quadtree variants. This enables the prediction of storage required by the representations and of the associated performance of algorithms that rely on them. The results are useful in two ways: 1. They reveal the properties of the various representations and permit their comparison using analytic, nonexperimental, criteria. Some of the results confirm previous analyses (e.g., that the storage requirement of the MX quadtree is proportional to the total lengths of the line segments). An important new result is that for a PMR and Bucket PMR quadtree with sufficiently high values of the splitting threshold (i.e., # 4) the number of nodes is proportional to the numberof line segments and is independent of the maximum depth of the tree. This provides a theoretical justification for ...
Phase changes in random point quadtrees
, 2005
"... Dedicated to the memory of ChingZong Wei (1949–2004) We show that a wide class of linear cost measures (such as the number of leaves) in random ddimensional point quadtrees undergo a change in limit laws: if the dimension d D 1;:::;8, then the limit law is normal; if d 9 then there is no convergenc ..."
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Dedicated to the memory of ChingZong Wei (1949–2004) We show that a wide class of linear cost measures (such as the number of leaves) in random ddimensional point quadtrees undergo a change in limit laws: if the dimension d D 1;:::;8, then the limit law is normal; if d 9 then there is no convergence to a fixed limit law. Stronger approximation results such as convergence rates and local limit theorems are also derived for the number of leaves, additional phase changes being unveiled. Our approach is new and very general, and also applicable to other classes of search trees. A brief discussion of Devroye’s gridtrees (covering mary search trees and quadtrees as special cases) is given. We also propose an efficient numerical procedure for computing the constants involved to high precision. Contents
Abstract
, 2000
"... The size of five triebased 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. S ..."
Abstract
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The size of five triebased 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 twodimensional, the trie is formed by interleaving the bits corresponding to the binary representation of the x and y coordinates of the underlying space and then testing two bits at each iteration. The result of this formulation yields a class of representations that are referred to as quadtrie variants, although they have been traditionally referred to as quadtree variants. The analysis differs from prior work in that it uses a detailed explicit model of the image instead of relying on modeling the branching process represented by the tree and leaving the underlying image unspecified. The analysis provides analytic expressions and bounds on the expected size of these quadtree variants. This enables the prediction of storage required by the representations and of the associated performance of algorithms that rely on them. The results are useful in two ways: 1. They reveal the properties of the various representations and permit their comparison using analytic, nonexperimental, criteria. Some of the results confirm previous analyses (e.g., that the storage requirement of the MX quadtree is proportional to the total lengths of the line segments). An important new result is that for a PMR and Bucket PMR quadtree with sufficiently high values of the splitting threshold (i.e., ¡ 4) the number of nodes is