arXiv subjects

Sean Cleary

Publications and source records attributed to Sean Cleary.

At least 19 recordsLinked to original sources

arXiv2025

The GFB Tree and Tree Imbalance Indices

Tree balance plays an important role in various research areas in phylogenetics and computer science. Typically, it is measured with the help of a balance index or imbalance index. There are more than 25 such indices available, recently surveyed in a book by Fischer et al. They are used to rank rooted binary trees on a scale from the most balanced to the least balanced. We show that a wide range of subtree-size based measures satisfying concavity and monotonicity conditions are minimized by the complete or greedy-from-the-bottom (GFB) tree and maximized by the caterpillar tree, yielding an infinitely large family of distinct new imbalance indices. Answering an open question from the literature, we show that one such established measure, the $\widehat{s}$-shape statistic, has the GFB tree as its unique minimizer. We also provide an alternative characterization of GFB trees, showing that they are equivalent to complete trees, which arise in different contexts. We give asymptotic bounds on the expected $\widehat{s}$-shape statistic under the uniform and Yule-Harding distributions of trees, and answer questions for the related $Q$-shape statistic as well.

q-bio.PE↗

arXiv2024

Grid diagrams for higher-dimensional Thompson's groups

We describe standard forms for elements of the higher-dimensional Thompson groups $nV$ arising from gridding subdivision processes. These processes lead to standard normal form descriptions for elements in these groups, and sizes of these standard forms estimate the word length with respect to finite generating sets. These gridded forms lead to standard algebraic descriptions as well, with respect to the both infinite and finite generating sets for these groups.

math.GR↗

arXiv2020

Elements with north-south dynamics and free subgroups have positive density in Thompson's groups T and V

We show that the density of elements in Thompson's groups T and V which have north-south dynamics acting on the circle is positive with respect to a stratification in terms of size. We show that the fraction of element pairs which generate a free subgroup of rank two is positive in both groups with respect to a stratification based on diagram size, which shows that the associated subgroup spectra of $T$ and $V$ contain free groups.

math.GR↗

arXiv2020

Distributions of restricted rotation distances

Rotation distances measure the differences in structure between rooted ordered binary trees. The one-dimensional skeleta of associahedra are rotation graphs, where two vertices representing trees are connected by an edge if they differ by a single rotation. There are no known efficient algorithms to compute rotation distance between trees and thus distances in rotation graphs. Limiting the allowed locations of where rotations are permitted gives rise to a number of notions of restricted rotation distances. Allowing rotations at a minimal such set of locations gives restricted rotation distance. There are linear-time algorithms to compute restricted rotation distance, where there are only two permitted locations for rotations to occur. The associated restricted rotation graph has an efficient distance algorithm. There are linear upper and lower bounds on restricted rotation distance with respect to the sizes of the reduced tree pairs. Here, we experimentally investigate the expected restricted rotation distance between two trees selected at random of increasing size and find that it lies typically in a narrow band well within the earlier proven linear upper and lower bounds.

cs.DS↗

arXiv2020

Counting difficult tree pairs with respect to the rotation distance problem

Rotation distance between rooted binary trees is the minimum number of simple rotations needed to transform one tree into the other. Computing the rotation distance between a pair of rooted trees can be quickly reduced in cases where there is a common edge between the trees, or where a single rotation introduces a common edge. Tree pairs which do not have such a reduction are difficult tree pairs, where there is no generally known first step. Here, we describe efforts to count and estimate the number of such difficult tree pairs, and find that the fraction decreases exponentially fast toward zero. We also describe how knowing the number of distinct instances of the rotation distance problem is a helpful factor in making the computations more feasible.

cs.DS↗

arXiv2020

An efficient sampling algorithm for difficult tree pairs

It is an open question whether there exists a polynomial-time algorithm for computing the rotation distances between pairs of extended ordered binary trees. The problem of computing the rotation distance between an arbitrary pair of trees, (S, T), can be efficiently reduced to the problem of computing the rotation distance between a difficult pair of trees (S', T'), where there is no known first step which is guaranteed to be the beginning of a minimal length path. Of interest, therefore, is how to sample such difficult pairs of trees of a fixed size. We show that it is possible to do so efficiently, and present such an algorithm that runs in time $O(n^4)$.

cs.DS↗

arXiv2017

Some Distributions on Finite Rooted Binary Trees

We introduce some natural families of distributions on rooted binary ranked plane trees with a view toward unifying ideas from various fields, including macroevolution, epidemiology, computational group theory, search algorithms and other fields. In the process we introduce the notions of split-exchangeability and plane-invariance of a general Markov splitting model in order to readily obtain probabilities over various equivalence classes of trees that arise in statistics, phylogenetics, epidemiology and group theory.

math.CO↗

arXiv2015

Edge conflicts do not determine geodesics in the associahedron

There are no known efficient algorithms to calculate distance in the one-skeleta of associahedra, a problem that is equivalent to finding rotation distance between rooted binary trees or the flip distance between polygonal triangulations. One measure of the difference between trees is the number of conflicting edge pairs, and a natural way of trying to find short paths is to minimize successively this number of conflicting edge pairs using flip operations in the corresponding triangulations. We describe examples that show that the number of such conflicts does not always decrease along geodesics. Thus, a greedy algorithm that always chooses a transformation that reduces conflicts will not produce a geodesic in all cases. Further, for any specified amount, there are examples of pairs of all large sizes showing that the number of conflicts can increase by that amount along any geodesic between the pairs.

math.CO↗

arXiv2015

Undistorted embeddings of metabelian groups of finite Pr\"ufer rank

General arguments of Baumslag and Bieri guarantee that any metabelian group of finite Pr\"ufer rank can be embedded in a metabelian constructible group. Here, we consider the metric behavior of a rich class of examples and analyze the distortions of specific embeddings.

math.GR↗

arXiv2014

Commensurations and Metric Properties of Houghton's Groups

We describe the automorphism groups and the abstract commensurators of Houghton's groups. Then we give sharp estimates for the word metric of these groups and deduce that the commensurators embed into the corresponding quasi-isometry groups. As a further consequence, we obtain that the Houghton group on two rays is at least quadratically distorted in those with three or more rays.

math.GR↗

arXiv2014

Obstructions for subgroups of Thompson's group $V$

Thompson's group $V$ has a rich variety of subgroups, containing all finite groups, all finitely generated free groups and all finitely generated abelian groups, the finitary permutation group of a countable set, as well as many wreath products and other families of groups. Here, we describe some obstructions for a given group to be a subgroup of $V$.

math.GR↗

arXiv2013

Addendum to "Commensurations and Subgroups of Finite Index of Thompson's Group F"

We show that the abstract commensurator of Thompson's group F is composed of four building blocks: two isomorphism types of simple groups, the multiplicative group of the positive rationals and a cyclic group of order two. The main result establishes the simplicity of a certain group of piecewise linear homeomorphisms of the real line.

math.GR↗

arXiv2011

The automorphism group of Thompson's group F: subgroups and metric properties

We describe some of the geometric properties of the automorphism group Aut(F) of Thompson's group F. We give realizations of Aut(F) geometrically via periodic tree pair diagrams, which lead to natural presentations and give effective methods for estimating the word length of elements. We study some natural subgroups of Aut(F) and their metric properties. In particular, we show that the subgroup of inner automorphisms of F is at least quadratically distorted in Aut(F), whereas other subgroups of Aut(F) isomorphic to F are undistorted.

math.GR↗

arXiv2009

Tame combing and almost convexity conditions

We give the first examples of groups which admit a tame combing with linear radial tameness function with respect to any choice of finite presentation, but which are not minimally almost convex on a standard generating set. Namely, we explicitly construct such combings for Thompson's group F and the Baumslag-Solitar groups BS(1, p) with p \ge 3. In order to make this construction for Thompson's group F, we significantly expand the understanding of the Cayley complex of this group with respect to the standard finite presentation. In particular we describe a quasigeodesic set of normal forms and combinatorially classify the arrangements of 2-cells adjacent to edges that do not lie on normal form paths.

math.GR↗

arXiv2009

Rotation Distance is Fixed-Parameter Tractable

Rotation distance between trees measures the number of simple operations it takes to transform one tree into another. There are no known polynomial-time algorithms for computing rotation distance. In the case of ordered rooted trees, we show that the rotation distance between two ordered trees is fixed-parameter tractable, in the parameter, k, the rotation distance. The proof relies on the kernalization of the initial trees to trees with size bounded by 7k.

cs.DS↗

arXiv2009

A Linear-Time Approximation Algorithm for Rotation Distance

Rotation distance between rooted binary trees measures the number of simple operations it takes to transform one tree into another. There are no known polynomial-time algorithms for computing rotation distance. We give an efficient, linear-time approximation algorithm, which estimates the rotation distance, within a provable factor of 2, between ordered rooted binary trees. .

cs.DS↗

arXiv2008

Metric properties of higher-dimensional Thompson's groups

Higher-dimensional Thompson's groups nV are finitely presented groups described by Brin which generalize dyadic self-maps of the unit interval to dyadic self-maps of n-dimensional unit cubes. We describe some of the metric properties of higher-dimensional Thompson's groups. We give descriptions of elements based upon tree-pair diagrams and give upper and lower bounds for word length in terms of the size of the diagrams. Though these upper and lower bounds are somewhat separated, we show that there are elements realizing the lower bounds and that the fraction of elements which are close to the upper bound converges to 1, showing that the bounds are optimal and that the upper bound is generically achieved.

math.GR↗

arXiv2007

Random subgroups of Thompson's group $F$

We consider random subgroups of Thompson's group $F$ with respect to two natural stratifications of the set of all $k$ generator subgroups. We find that the isomorphism classes of subgroups which occur with positive density are not the same for the two stratifications. We give the first known examples of {\em persistent} subgroups, whose isomorphism classes occur with positive density within the set of $k$-generator subgroups, for all sufficiently large $k$. Additionally, Thompson's group provides the first example of a group without a generic isomorphism class of subgroup. Elements of $F$ are represented uniquely by reduced pairs of finite rooted binary trees. We compute the asymptotic growth rate and a generating function for the number of reduced pairs of trees, which we show is D-finite and not algebraic. We then use the asymptotic growth to prove our density results.

math.GR↗