scholarly journals Limit Theorems for Depths and Distances in Weighted Random B-Ary Recursive Trees

2011 ◽  
Vol 48 (4) ◽  
pp. 1060-1080 ◽  
Author(s):  
Götz Olaf Munsonius ◽  
Ludger Rüschendorf
Keyword(s):  
Path Length ◽  
Limit Theorems ◽  
Wiener Index ◽  
Random Trees ◽  
Random Node ◽  
Recursive Trees

Limit theorems are established for some functionals of the distances between two nodes in weighted random b-ary recursive trees. We consider the depth of the nth node and of a random node, the distance between two random nodes, the internal path length, and the Wiener index. As an application, these limit results imply, by an imbedding argument, corresponding limit theorems for further classes of random trees: plane-oriented recursive trees and random linear recursive trees.

scholarly journals Limit Theorems for Depths and Distances in Weighted Random B-Ary Recursive Trees

2011 ◽  
Vol 48 (04) ◽  
pp. 1060-1080 ◽  
Author(s):  
Götz Olaf Munsonius ◽  
Ludger Rüschendorf
Keyword(s):  
Path Length ◽  
Limit Theorems ◽  
Wiener Index ◽  
Random Trees ◽  
Random Node ◽  
Recursive Trees

Limit theorems are established for some functionals of the distances between two nodes in weighted randomb-ary recursive trees. We consider the depth of thenth node and of a random node, the distance between two random nodes, the internal path length, and the Wiener index. As an application, these limit results imply, by an imbedding argument, corresponding limit theorems for further classes of random trees: plane-oriented recursive trees and random linear recursive trees.

scholarly journals The Wiener Index of Random Trees

2002 ◽  
Vol 11 (6) ◽  
pp. 587-597 ◽  
Author(s):  
RALPH NEININGER
Keyword(s):  
Path Length ◽  
Probabilistic Models ◽  
Wiener Index ◽  
Binary Search ◽  
Random Trees ◽  
Binary Search Trees ◽  
Search Trees ◽  
Limit Laws ◽  
Recursive Trees ◽  
Asymptotic Correlations

The Wiener index is analysed for random recursive trees and random binary search trees in uniform probabilistic models. We obtain expectations, asymptotics for the variances, and limit laws for this parameter. The limit distributions are characterized as the projections of bivariate measures that satisfy certain fixed point equations. Covariances, asymptotic correlations, and bivariate limit laws for the Wiener index and the internal path length are given.

scholarly journals On Tail Bounds for Random Recursive Trees

2012 ◽  
Vol 49 (2) ◽  
pp. 566-581 ◽  
Author(s):  
Götz Olaf Munsonius
Keyword(s):  
Analysis Of Algorithms ◽  
Wiener Index ◽  
Binary Search ◽  
Random Trees ◽  
Stochastic Domination ◽  
Lower Tail ◽  
Probabilistic Analysis Of Algorithms ◽  
The Laplace Transform ◽  
Recursive Trees ◽  
Tail Bound

We consider a multivariate distributional recursion of sum type, as arises in the probabilistic analysis of algorithms and random trees. We prove an upper tail bound for the solution using Chernoff's bounding technique by estimating the Laplace transform. The problem is traced back to the corresponding problem for binary search trees by stochastic domination. The result obtained is applied to the internal path length and Wiener index of random b-ary recursive trees with weighted edges and random linear recursive trees. Finally, lower tail bounds for the Wiener index of these trees are given.

scholarly journals On Tail Bounds for Random Recursive Trees

2012 ◽  
Vol 49 (02) ◽  
pp. 566-581 ◽  
Author(s):  
Götz Olaf Munsonius
Keyword(s):  
Analysis Of Algorithms ◽  
Wiener Index ◽  
Binary Search ◽  
Random Trees ◽  
Stochastic Domination ◽  
Lower Tail ◽  
Probabilistic Analysis Of Algorithms ◽  
The Laplace Transform ◽  
Recursive Trees ◽  
Tail Bound

We consider a multivariate distributional recursion of sum type, as arises in the probabilistic analysis of algorithms and random trees. We prove an upper tail bound for the solution using Chernoff's bounding technique by estimating the Laplace transform. The problem is traced back to the corresponding problem for binary search trees by stochastic domination. The result obtained is applied to the internal path length and Wiener index of randomb-ary recursive trees with weighted edges and random linear recursive trees. Finally, lower tail bounds for the Wiener index of these trees are given.

scholarly journals Profiles of Random Trees: Limit Theorems for Random Recursive Trees and Binary Search Trees

Algorithmica ◽  
2006 ◽  
Vol 46 (3-4) ◽  
pp. 367-407 ◽  
Author(s):  
Michael Fuchs ◽  
Hsien-Kuei Hwang ◽  
Ralph Neininger
Keyword(s):  
Limit Theorems ◽  
Binary Search ◽  
Random Trees ◽  
Binary Search Trees ◽  
Search Trees ◽  
Recursive Trees

Tangent bundle RRT: A randomized algorithm for constrained motion planning

Robotica ◽  
2014 ◽  
Vol 34 (1) ◽  
pp. 202-225 ◽  
Author(s):  
Beobkyoon Kim ◽  
Terry Taewoong Um ◽  
Chansu Suh ◽  
F. C. Park
Keyword(s):  
Motion Planning ◽  
Configuration Space ◽  
Tangent Bundle ◽  
Numerical Experiments ◽  
Randomized Algorithm ◽  
Random Trees ◽  
Constrained Motion ◽  
Wide Range ◽  
Random Node ◽  
Planning Problems

SUMMARYThe Tangent Bundle Rapidly Exploring Random Tree (TB-RRT) is an algorithm for planning robot motions on curved configuration space manifolds, in which the key idea is to construct random trees not on the manifold itself, but on tangent bundle approximations to the manifold. Curvature-based methods are developed for constructing tangent bundle approximations, and procedures for random node generation and bidirectional tree extension are developed that significantly reduce the number of projections to the manifold. Extensive numerical experiments for a wide range of planning problems demonstrate the computational advantages of the TB-RRT algorithm over existing constrained path planning algorithms.

DEPTH AND PATH LENGTH OF HEAP ORDERED TREES

1996 ◽  
Vol 07 (03) ◽  
pp. 293-299 ◽  
Author(s):  
HELMUT PRODINGER
Keyword(s):  
General Theory ◽  
Path Length ◽  
Ad Hoc ◽  
Point Of View ◽  
Plane Tree ◽  
Ordered Trees ◽  
Natural Concept ◽  
Random Node ◽  
Ordered Tree

A heap ordered tree of size n is a planted plane tree together with a bijection from the nodes to the set {1,…,n} which is monotonically increasing when going from the root to the leaves. In a recent paper by Chen and Ni, the expectation and the variance of the depth of a random node in a random heap ordered tree of size n was considered. Here, we give additional results concerning level polynomials. From a historical point of view, we trace these trees back to an earlier paper by Prodinger and Urbanek and find formulae that are proved in the paper by Chen and Ni by ad hoc computations already in a classic book by Greene and Knuth. Also, we mention that a paper by Bergeron, Flajolet and Salvy develops a more general theory of increasing trees, based on simply generated families of trees. Furthermore we consider the path length which is a natural concept when dealing with the depth. Expectation and variance are obtained, both explicitly and asymptotically.

scholarly journals Central limit theorems for generalized Pólya urn models

2006 ◽  
Vol 43 (4) ◽  
pp. 938-951 ◽  
Author(s):  
I. Higueras ◽  
J. Moler ◽  
F. Plo ◽  
M. San Miguel
Keyword(s):  
Covariance Matrix ◽  
Central Limit ◽  
Limit Theorems ◽  
Lyapunov Equation ◽  
Central Limit Theorems ◽  
Random Trees ◽  
Urn Models ◽  
Pólya Urn ◽  
Polya Urn ◽  
Generalized Pólya Urn

In this paper we obtain central limit theorems for generalized Pólya urn models with L ≥ 2 colors where one out of K different replacements (actions) is applied randomly at each step. Each possible action constitutes a row of the replacement matrix, which can be nonsquare and random. The actions are chosen following a probability distribution given by an arbitrary function of the proportions of the balls of different colors present in the urn. Moreover, under the same hypotheses it is proved that the covariance matrix of the asymptotic distribution is the solution of a Lyapunov equation, and a procedure is given to obtain the covariance matrix in an explicit form. Some applications of these results to random trees and adaptive designs in clinical trials are also presented.

scholarly journals Central limit theorems for generalized Pólya urn models

2006 ◽  
Vol 43 (04) ◽  
pp. 938-951
Author(s):  
I. Higueras ◽  
J. Moler ◽  
F. Plo ◽  
M. San Miguel
Keyword(s):  
Covariance Matrix ◽  
Central Limit ◽  
Limit Theorems ◽  
Lyapunov Equation ◽  
Central Limit Theorems ◽  
Random Trees ◽  
Urn Models ◽  
Pólya Urn ◽  
Polya Urn ◽  
Generalized Pólya Urn

In this paper we obtain central limit theorems for generalized Pólya urn models with L ≥ 2 colors where one out of K different replacements (actions) is applied randomly at each step. Each possible action constitutes a row of the replacement matrix, which can be nonsquare and random. The actions are chosen following a probability distribution given by an arbitrary function of the proportions of the balls of different colors present in the urn. Moreover, under the same hypotheses it is proved that the covariance matrix of the asymptotic distribution is the solution of a Lyapunov equation, and a procedure is given to obtain the covariance matrix in an explicit form. Some applications of these results to random trees and adaptive designs in clinical trials are also presented.

scholarly journals Deterministic Edge Weights in Increasing Tree Families

2009 ◽  
Vol 19 (1) ◽  
pp. 99-119
Author(s):  
MARKUS KUBA ◽  
STEPHAN WAGNER
Keyword(s):  
Normal Limit ◽  
Random Trees ◽  
Edge Weight ◽  
Combinatorial Approach ◽  
Related Parameter ◽  
Scale Free ◽  
Recursive Tree ◽  
Combinatorial Model ◽  
Recursive Trees

In this work we study edge weights for two specific families of increasing trees, which include binary increasing trees and plane-oriented recursive trees as special instances, where plane-oriented recursive trees serve as a combinatorial model of scale-free random trees given by the m = 1 case of the Barabási–Albert model. An edge e = (k, l), connecting the nodes labelled k and l, respectively, in an increasing tree, is associated with the weight we = |k − l|. We are interested in the distribution of the number of edges with a fixed edge weight j in a random generalized plane-oriented recursive tree or random d-ary increasing tree. We provide exact formulas for expectation and variance and prove a normal limit law for this quantity. A combinatorial approach is also presented and applied to a related parameter, the maximum edge weight.

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