Results 1  10
of
21
Universal Limit Laws for Depths in Random Trees
 SIAM Journal on Computing
, 1998
"... Random binary search trees, bary search trees, medianof(2k+1) trees, quadtrees, simplex trees, tries, and digital search trees are special cases of random split trees. For these trees, we o#er a universal law of large numbers and a limit law for the depth of the last inserted point, as well as a ..."
Abstract

Cited by 56 (8 self)
 Add to MetaCart
Random binary search trees, bary search trees, medianof(2k+1) trees, quadtrees, simplex trees, tries, and digital search trees are special cases of random split trees. For these trees, we o#er a universal law of large numbers and a limit law for the depth of the last inserted point, as well as a law of large numbers for the height.
On a multivariate contraction method for random recursive structures with applications to Quicksort
, 2001
"... The contraction method for recursive algorithms is extended to the multivariate analysis of vectors of parameters of recursive structures and algorithms. We prove a general multivariate limit law which also leads to an approach to asymptotic covariances and correlations of the parameters. As an appl ..."
Abstract

Cited by 35 (16 self)
 Add to MetaCart
The contraction method for recursive algorithms is extended to the multivariate analysis of vectors of parameters of recursive structures and algorithms. We prove a general multivariate limit law which also leads to an approach to asymptotic covariances and correlations of the parameters. As an application the asymptotic correlations and a bivariate limit law for the number of key comparisons and exchanges of medianof(2t + 1) Quicksort is given. Moreover, for the Quicksort programs analyzed by Sedgewick the exact order of the standard deviation and a limit law follow, considering all the parameters counted by Sedgewick.
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 ..."
Abstract

Cited by 27 (3 self)
 Add to MetaCart
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.
On the internal path length of ddimensional quad trees
, 1999
"... It is proved that the internal path length of a d–dimensional quad tree after normalization converges in distribution. The limiting distribution is characterized as a fixed point of a random affine operator. We obtain convergence of all moments and of the Laplace transforms. The moments of the limit ..."
Abstract

Cited by 21 (10 self)
 Add to MetaCart
It is proved that the internal path length of a d–dimensional quad tree after normalization converges in distribution. The limiting distribution is characterized as a fixed point of a random affine operator. We obtain convergence of all moments and of the Laplace transforms. The moments of the limiting distribution can be evaluated from the recursion and lead to first order asymptotics for the moments of the internal path lengths. The analysis is based on the contraction method. In the final part of the paper we state similar results for general split tree models if the expectation of the path length has a similar expansion as in the case of quad trees. This applies in particular to the mary search trees.
The Average Case Analysis of Algorithms: Multivariate Asymptotics and Limit Distributions
, 1997
"... This report is part of a series whose aim is to present in a synthetic way the major methods of "analytic combinatorics" needed in the averagecase analysis of algorithms. It develops a general approach to the distributional analysis of parameters of elementary combinatorial structures li ..."
Abstract

Cited by 16 (1 self)
 Add to MetaCart
This report is part of a series whose aim is to present in a synthetic way the major methods of "analytic combinatorics" needed in the averagecase analysis of algorithms. It develops a general approach to the distributional analysis of parameters of elementary combinatorial structures like strings, trees, graphs, permutations, and so on. The methods are essentially analytic and relie on multivariate generating functions, singularity analysis, and continuity theorems. The limit laws that are derived mostly belong to the Gaussian, Poisson, or geometric type.
Partial match queries in random quadtrees
 SIAM J. Comput
, 2003
"... 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 i ..."
Abstract

Cited by 8 (1 self)
 Add to MetaCart
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. Key words. Quadtrees, partial match queries, binomial transform, Mellin transform, Euler transform, Rice’s integral.
Squarish kd trees
 SIAM JOURNAL ON COMPUTING
, 2000
"... We modify the kd tree on [0, 1] d by always cutting the longest edge instead of rotating through the coordinates. This modification makes the expected time behavior of lowerdimensional partial match queries behave as for perfectly balanced complete kd trees on n nodes. This is in contrast to a ..."
Abstract

Cited by 8 (3 self)
 Add to MetaCart
We modify the kd tree on [0, 1] d by always cutting the longest edge instead of rotating through the coordinates. This modification makes the expected time behavior of lowerdimensional partial match queries behave as for perfectly balanced complete kd trees on n nodes. This is in contrast to a result of Flajolet and Puech, who proved that for (standard) random kd trees with cuts that rotate among the coordinate axes, the expected time behavior was much worse than for balanced complete kd trees. We also provide results for range searching and nearest neighbor search for our trees.