12 research outputs found
Random-bit optimal uniform sampling for rooted planar trees with given sequence of degrees and Applications
In this paper, we redesign and simplify an algorithm due to Remy et al. for
the generation of rooted planar trees that satisfies a given partition of
degrees. This new version is now optimal in terms of random bit complexity, up
to a multiplicative constant. We then apply a natural process
"simulate-guess-and-proof" to analyze the height of a random Motzkin in
function of its frequency of unary nodes. When the number of unary nodes
dominates, we prove some unconventional height phenomenon (i.e. outside the
universal square root behaviour.)Comment: 19 page
Exact-size Sampling for Motzkin Trees in Linear Time via Boltzmann Samplers and Holonomic Specification
International audienceBoltzmann samplers are a kind of random samplers; in 2004, Duchon, Flajolet, Louchard and Schaeffer showed that given a combinatorial class and a combinatorial specification for that class, one can automatically build a Boltzmann sampler. In this paper, we introduce a Boltzmann sampler for Motzkin trees built from a holonomic specification, that is, a specification that uses the pointing operator. This sampler is inspired by Rémy's algorithm on binary trees. We show that our algorithm gives an exact size sampler with a linear time and space complexity in average
Holonomic equations and efficient random generation of binary trees
Holonomic equations are recursive equations which allow computingefficiently
numbers of combinatoric objects. R{\'e}my showed that theholonomic equation
associated with binary trees yields an efficientlinear random generator of
binary trees. I extend this paradigm toMotzkin trees and Schr{\"o}der trees and
show that despite slightdifferences my algorithm that generates random
Schr{\"o}der trees has linearexpected complexity and my algorithm that
generates Motzkin trees is inO(n) expected complexity, only if we can implement
a specific oraclewith a O(1) complexity. For Motzkin trees, I propose a
solution whichworks well for realistic values (up to size ten millions) and
yields anefficient algorithm
Statistical tests for large tree-structured data
We develop a general statistical framework for the analysis and inference of large tree-structured data, with a focus on developing asymptotic goodness-of-fit tests. We first propose a consistent statistical model for binary trees, from which we develop a class of invariant tests. Using the model for binary trees, we then construct tests for general trees by using the distributional properties of the Continuum Random Tree, which arises as the invariant limit for a broad class of models for tree-structured data based on conditioned Galton–Watson processes. The test statistics for the goodness-of-fit tests are simple to compute and are asymptotically distributed as χ2 and F random variables. We illustrate our methods on an important application of detecting tumour heterogeneity in brain cancer. We use a novel approach with tree-based representations of magnetic resonance images and employ the developed tests to ascertain tumor heterogeneity between two groups of patients
Scaling limits of slim and fat trees
We consider Galton--Watson trees conditioned on both the total number of
vertices and the number of leaves . The focus is on the case in which
both and grow to infinity and , with . Assuming the exponential decay of the offspring distribution, we show
that the rescaled random tree converges in distribution to Aldous' Continuum
Random Tree with respect to the Gromov--Hausdorff topology. The scaling depends
on a parameter which we calculate explicitly. Additionally, we
compute the limit for the degree sequences of these random trees.Comment: 37 pages, 2 figure