1,395 research outputs found
Limiting distributions for additive functionals on Catalan trees
Additive tree functionals represent the cost of many divide-and-conquer
algorithms. We derive the limiting distribution of the additive functionals
induced by toll functions of the form (a) n^\alpha when \alpha > 0 and (b) log
n (the so-called shape functional) on uniformly distributed binary search
trees, sometimes called Catalan trees. The Gaussian law obtained in the latter
case complements the central limit theorem for the shape functional under the
random permutation model. Our results give rise to an apparently new family of
distributions containing the Airy distribution (\alpha = 1) and the normal
distribution [case (b), and case (a) as ]. The main
theoretical tools employed are recent results relating asymptotics of the
generating functions of sequences to those of their Hadamard product, and the
method of moments.Comment: 30 pages, 4 figures. Version 2 adds background information on
singularity analysis and streamlines the presentatio
Transfer Theorems and Asymptotic Distributional Results for m-ary Search Trees
We 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
[roughly, ] we have asymptotic normality if and
typically periodic behavior if ; (ii) for moderate toll sequences
[roughly, but ] we have convergence to
non-normal distributions if (where ) and typically
periodic behavior if ; and (iii) for large toll sequences
[roughly, ] we have convergence to non-normal distributions
for all values of m.Comment: 35 pages, 1 figure. Version 2 consists of expansion and rearragement
of the introductory material to aid exposition and the shortening of
Appendices A and B.
Limit Laws for Functions of Fringe trees for Binary Search Trees and Recursive Trees
We prove limit theorems for sums of functions of subtrees of binary search
trees and random recursive trees. In particular, we give simple new proofs of
the fact that the number of fringe trees of size in the binary search
tree and the random recursive tree (of total size ) asymptotically has a
Poisson distribution if , and that the distribution is
asymptotically normal for . Furthermore, we prove similar
results for the number of subtrees of size with some required property , for example the number of copies of a certain fixed subtree . Using
the Cram\'er-Wold device, we show also that these random numbers for different
fixed subtrees converge jointly to a multivariate normal distribution. As an
application of the general results, we obtain a normal limit law for the number
of -protected nodes in a binary search tree or random recursive tree.
The proofs use a new version of a representation by Devroye, and Stein's
method (for both normal and Poisson approximation) together with certain
couplings
Simply Generated Trees, B-series and Wigner Processes
We consider simply generated trees and study multiplicative functions on
rooted plane trees. We show that the associated generating functions satisfy
differential equations or difference equations. Our approach considers B-series
from Butcher's theory, the generating functions are seen as generalized
Runge-Kutta methodsComment: 19 pages, 1 figur
Asymptotic Normality of Almost Local Functionals in Conditioned Galton-Watson Trees
An additive functional of a rooted tree is a functional that can be calculated recursively as the sum of the values of the functional over the branches, plus a certain toll function. Janson recently proved a central limit theorem for additive functionals of conditioned Galton-Watson trees under the assumption that the toll function is local, i.e. only depends on a fixed neighbourhood of the root. We extend his result to functionals that are almost local, thus covering a wider range of functionals. Our main result is illustrated by two explicit examples: the (logarithm of) the number of matchings, and a functional stemming from a tree reduction process that was studied by Hackl, Heuberger, Kropf, and Prodinger
Central limit theorems for fringe trees in patricia tries
We give theorems about asymptotic normality of general additive functionals
on patricia tries in an i.i.d. setting, derived from results on tries by Janson
(2022). These theorems are applied to show asymptotic normality of the
distribution of random fringe trees in patricia tries. Formulas for asymptotic
mean and variance are given. The proportion of fringe trees with keys is
asymptotically, ignoring oscillations, given by with
the source entropy , an entropy-like constant , that is in the binary
case, and an exponentially decreasing function . Another application
gives asymptotic normality of the independence number
On the exponential functional of Markov Additive Processes, and applications to multi-type self-similar fragmentation processes and trees
A Markov Additive Process is a bi-variate Markov process
which should be thought of as a
multi-type L\'evy process: the second component is a Markov chain on a
finite space , and the first component behaves locally as
a L\'evy process, with local dynamics depending on . In the
subordinator-like case where is nondecreasing, we establish several
results concerning the moments of and of its exponential functional
extending the work of Carmona
et al., and Bertoin and Yor.
We then apply these results to the study of multi-type self-similar
fragmentation processes: these are self-similar analogues of Bertoin's
homogeneous multi-type fragmentation processes Notably, we encode the genealogy
of the process in a tree, and under some Malthusian hypotheses, compute its
Hausdorff dimension in a generalisation of our previous work.Comment: Minor corrections and typo
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