Internal Path Length Research Articles

Proper generating trees and their internal path length

We find the generating function counting the total internal path length of any proper generating tree. This function is expressed in terms of the functions ( d ( t ) , h ( t ) ) defining the associated proper Riordan array. This result is important in the theory of Riordan arrays and has several combinatorial interpretations.

Tail Bounds for the Wiener Index of Random Trees

Upper and lower bounds for the tail probabilities of the Wiener index of random binary search trees are given. For upper bounds the moment generating function of the vector of Wiener index and internal path length is estimated. For the lower bounds a tree class with sufficiently large probability and atypically large Wiener index is constructed. The methods are also applicable to related random search trees.

Note on the weighted internal path length of b-ary trees

Analysis of Algorithms In a recent paper Broutin and Devroye (2005) have studied the height of a class of edge-weighted random trees.This is a class of trees growing in continuous time which includes many wellknown trees as examples. In this paper we derive a limit theorem for the internal path length for this class of trees.For the proof we extend a limit theorem in Neininger and Rüschendorf (2004) to recursive sequences of random variables with continuous time parameter.

Transfer theorems and asymptotic distributional results for m‐ary search trees

AbstractWe derive asymptotics of moments and identify limiting distributions, under the random permutation model on m‐ary search trees, for functionals that satisfy recurrence relations of a simple additive form. Many important functionals including the space requirement, internal path length, and the so‐called shape functional fall under this framework. The approach is based on establishing transfer theorems that link the order of growth of the input into a particular (deterministic) recurrence to the order of growth of the output. The transfer theorems are used in conjunction with the method of moments to establish limit laws. It is shown that: (i) for small toll sequences (tn) [roughly, tn = O(n1/2)] we have asymptotic normality if m ≤ 26 and typically periodic behavior if m ≥ 27; (ii) for moderate toll sequences [roughly, tn = ω(n1/2) but tn = o(n)] we have convergence to nonnormal distributions if m ≤ m0 (where m0 ≥ 26) and typically periodic behavior if m ≥ m0 + 1; and (iii) for large toll sequences [roughly, tn = ω(n)] we have convergence to nonnormal distributions for all values of m. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2005

One-sided variations on binary search trees

We investigate incomplete one-sided variants of binary search trees. The (normed) size of each variant is studied, and convergence to a Gaussian law is proved in each case by asymptotically solving recurrences. These variations are also discussed within the scope of the contraction method with degenerate limit equations. In an incomplete tree the size determines most other parameters of interest, such as the height and the internal path length.

One-sided variations on interval trees

The binary interval tree is a random structure that underlies interval division and parking problems. Five incomplete one-sided variants of binary interval trees are considered, providing additional flavors and variations on the main applications. The size of each variant is studied, and a Gaussian tendency is proved in each case via an analytic approach. Differential equations on half scale and delayed differential equations arise and can be solved asymptotically by local expansions and Tauberian theorems. Unlike the binary case, in an incomplete interval tree the size determines most other parameters of interest, such as the height or the internal path length.

One-sided variations on interval trees

The binary interval tree is a random structure that underlies interval division and parking problems. Five incomplete one-sided variants of binary interval trees are considered, providing additional flavors and variations on the main applications. The size of each variant is studied, and a Gaussian tendency is proved in each case via an analytic approach. Differential equations on half scale and delayed differential equations arise and can be solved asymptotically by local expansions and Tauberian theorems. Unlike the binary case, in an incomplete interval tree the size determines most other parameters of interest, such as the height or the internal path length.

The Wiener Index of simply generated random trees

AbstractAsymptotics are obtained for the mean, variance, and higher moments as well as for the distribution of the Wiener index of a random tree from a simply generated family (or, equivalently, a critical Galton–Watson tree). We also establish a joint asymptotic distribution of the Wiener index and the internal path length, as well as asymptotics for the covariance and other mixed moments. The limit laws are described using functionals of a Brownian excursion. The methods include both Aldous' theory of the continuum random tree and analysis of generating functions. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 22: 337–358, 2003

The Wiener Index of Random Trees

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.

Laws of large numbers and tail inequalities for random tries and PATRICIA trees

We consider random tries and random PATRICIA trees constructed from n independent strings of symbols drawn from any distribution on any discrete space. If H n is the height of this tree, we show that H n/ E{H n} tends to one in probability. Additional tail inequalities are given for the height, depth, size, internal path length, and profile of these trees and ordinary tries that apply without any conditions on the string distributions—they need not even be identically distributed.

On binary search tree recursions with monomials as toll functions

We consider distributional recursions which appear in the study of random binary search trees with monomials as toll functions. This extends classical parameters as the internal path length in binary search trees. As our main results we derive asymptotic expansions for the moments of the random variables under consideration as well as limit laws and properties of the densities of the limit distributions. The analysis is based on the contraction method.

A Monte Carlo ray‐tracing model for scattering and polarization by large particles with complex shapes

We present a new model based on Monte Carlo ray tracing which simulates scattering and linear polarization by particles with arbitrary shapes and sizes much larger than the wavelength. The model, called S‐Scat, provides a powerful tool for exploring the relationship between actual physical properties of large particles and the single‐particle parameters used in multiple‐scattering radiative transfer models, and will be particularly valuable for studies of icy outer solar system surfaces. We describe the model algorithm and apply the model to examine absorption and scattering behavior of single, irregular particles as functions of particle size, shape, and optical constants. Single‐scattering albedos are investigated first, with results used to test the validity of the widely used equivalent slab model. Single‐scattering phase functions are examined next, along with possibilities of parameterization via simple analytic expressions. Finally, the behaviors of linear polarization functions are explored, along with internal path lengths and the spatial distribution of scattered light in close proximity to the particle.

A NEW WEIGHT BALANCED BINARY SEARCH TREE

We develop a new class of weight balanced binary search trees called β-balanced binary search trees (β-BBSTs). β-BBSTs are designed to have reduced internal path length. As a result, they are expected to exhibit good search time characteristics. Individual search, insert, and delete operations in an n node β-BBST take O( log n) time for [Formula: see text]. Experimental results comparing the performance of β-BBSTs, WB(α) trees, AVL-trees, red/black trees, treaps, deterministic skip lists and skip lists are presented. Two simplified versions of, β-BBSTs are also developed.

On the variance of the internal path length of generalized digital trees – The Mellin convolution approach

This paper studies a generalization of the internal path length of a random digital search tree to bucket trees, which are capable of storing up to b records per node. We show that under the assumption of the symmetric Bernoulli probabilistic model the expected path length of a tree built from N records is asymptotically N log 2N+(A+δ 1( log 2N))N and the variance is (C+ε 1( log 2N))N , where A and C depend on b only. The continuous functions δ 1 and ε 1 are periodic with mean 0 and period 1. The proofs are analytical and make use of generating functions, harmonic sums and the Mellin integral transform. An important and very general tool for the analysis is Mellin's convolution integral. These results and techniques are motivated by a paper by Flajolet and Richmond and by Kirschenhofer, Prodinger, and Szpankowski.

The Accuracy of Ground-Based Optical Interferometry Observations

AbstractThe problem of wide-angle astrometry via interferometry is to recover the two coordinates for each star from the observed delays. In the absence of atmospheric turbulence, the geometrical delay can be defined as: dG, ij(t) ≡ dj(t) ‒ di(t) = Bij(t)·ŝ0 ‒ Cij. Here the geometrical delay is the difference between the delay line lengths di and dj that is required to equalize the effective optical paths from the star to the point of beam combination via each of two apertures i and j. Bij(t) is the baseline between the apertures, ŝ0 is the star position, and Cij is the difference between the ‘fixed’ internal optical path lengths Ci and Cj within the instrument. In principle, sufficient delay measurements would allow solution for the baseline vectors and the delay constants, as well as the positions of the stars. However, the actual situation is greatly complicated by the presence of the atmosphere and the fact that neither the delay ‘constants’ Cij nor the baseline vectors are stable over time. The design of, and the analysis of the data from, any ground-based optical interferometer must overcome all three of these effects. The design, operation, and the analysis of data from the Navy Prototype Optical Interferometer (NPOI) are presented here as examples of how to overcome the effects of the atmosphere and the instrumental instabilities in order to achieve accurate wide-angle astrometry. The status of the implementation of these techniques at the NPOI is presented.

On the internal path length ofd-dimensional quad trees

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 m-ary search trees. ©1999 John Wiley & Sons, Inc. Random Struct. Alg., 5: 25–41, 1999

A methodology for plane tree enumeration

In this paper, we illustrate a method (called the ECO method) for enumerating some classes of combinatorial objects. By means of an operator, able to satisfy two particular conditions, we give some recursive descriptions of these subclasses. We use these descriptions to deduce the functional equations satisfied by the classes' generating functions. We then illustrate some applications of the method to the whole class of plane trees, to right-leafed trees and to tipaugmented trees, and determine their generating functions according to the number of their internal nodes, the number of their leaves, their right branch length and their internal path length.

Probabilistic analysis of some (un)directed animals

Using weak convergence theorems, we analyze stochastic properties of some parameters of directed or undirected animals. For directed column convex animals, with fixed large area, we obtain asymptotic distributions for the number of columns, the internal path length, the trajectories and the right width. For general directed animals, we derive asymptotic densities of the lower width and of some decomposition parameter. The limiting processes are given by Gaussian stochastic processes based on Brownian Motion or stochastic integrals on such.

Binary search trees constructed from nondistinct keys with/without specified probabilities

We investigate binary search trees formed from sequences of nondistinct keys under two models. In the first model, an input sequence is composed from elements of a given finite multiset and all possible sequences are equally-likely. In the second model, an input sequence is composed from n elements of a (possibly infinite) set, where each key has a specified probability; the n keys are independently chosen from the given set. Under both models, we shall derive general closed-form expressions for the expected values of the characteristic parameters defined on the corresponding binary search trees. These parameters include the (left, right) depth of a given key, the level of a given external node and the left (right) side or the internal (external) path length of a search tree. Furthermore, we find some nonobvious relations between these expected values. In some respects, the second model tends to the first model for large n. All results are illustrated by concrete examples sometimes showing unexpected phenomena.

Asymptotic behavior of the Lempel-Ziv parsing scheme and digital search trees

The Lempel-Ziv parsing scheme finds a wide range of applications, most notably in data compression and algorithms on words. It partitions a sequence of length n into variable phrases such that a new phrase is the shortest substring not seen in the past as a phase. The parameter of interest is the number Mn of phrases that one can construct from a sequence of length n. In this paper, for the memoryless source with unequal probabilities of symbols generation we derive the limiting distribution of Mn which turns out to be normal. This proves a long-standing open problem. In fact, to obtain this result we solved another open problem, namely, that of establishing the limiting distribution of the internal path length in a digital search tree. The latter is a consequence of an asymptotic solution of a multiplicative differential-functional equation often arising in the analysis of algorithms on words. Interestingly enough, our findings are proved by a combination of probabilistic techniques such as renewal equation and uniform integrability, and analytical techniques such as Mellin transform, differential-functional equations, de-Poissonization, and so forth. In concluding remarks we indicate a possibility of extending our results to Markovian models.