594,198 research outputs found
Pattern Formation on Trees
Networks having the geometry and the connectivity of trees are considered as
the spatial support of spatiotemporal dynamical processes. A tree is
characterized by two parameters: its ramification and its depth. The local
dynamics at the nodes of a tree is described by a nonlinear map, given rise to
a coupled map lattice system. The coupling is expressed by a matrix whose
eigenvectors constitute a basis on which spatial patterns on trees can be
expressed by linear combination. The spectrum of eigenvalues of the coupling
matrix exhibit a nonuniform distribution which manifest itself in the
bifurcation structure of the spatially synchronized modes. These models may
describe reaction-diffusion processes and several other phenomena occurring on
heterogeneous media with hierarchical structure.Comment: Submitted to Phys. Rev. E, 15 pages, 9 fig
Non-Contiguous Pattern Avoidance in Binary Trees
In this paper we consider the enumeration of binary trees avoiding
non-contiguous binary tree patterns. We begin by computing closed formulas for
the number of trees avoiding a single binary tree pattern with 4 or fewer
leaves and compare these results to analogous work for contiguous tree
patterns. Next, we give an explicit generating function that counts binary
trees avoiding a single non-contiguous tree pattern according to number of
leaves. In addition, we enumerate binary trees that simultaneously avoid more
than one tree pattern. Finally, we explore connections between pattern-avoiding
trees and pattern-avoiding permutations.Comment: 21 pages, 2 figures, 1 tabl
Pattern avoidance in binary trees
This paper considers the enumeration of trees avoiding a contiguous pattern.
We provide an algorithm for computing the generating function that counts
n-leaf binary trees avoiding a given binary tree pattern t. Equipped with this
counting mechanism, we study the analogue of Wilf equivalence in which two tree
patterns are equivalent if the respective n-leaf trees that avoid them are
equinumerous. We investigate the equivalence classes combinatorially. Toward
establishing bijective proofs of tree pattern equivalence, we develop a general
method of restructuring trees that conjecturally succeeds to produce an
explicit bijection for each pair of equivalent tree patterns.Comment: 19 pages, many images; published versio
Pattern avoidance in labelled trees
We discuss a new notion of pattern avoidance motivated by the operad theory:
pattern avoidance in planar labelled trees. It is a generalisation of various
types of consecutive pattern avoidance studied before: consecutive patterns in
words, permutations, coloured permutations etc. The notion of Wilf equivalence
for patterns in permutations admits a straightforward generalisation for (sets
of) tree patterns; we describe classes for trees with small numbers of leaves,
and give several bijections between trees avoiding pattern sets from the same
class. We also explain a few general results for tree pattern avoidance, both
for the exact and the asymptotic enumeration.Comment: 27 pages, corrected various misprints, added an appendix explaining
the operadic contex
Linear pattern matching on sparse suffix trees
Packing several characters into one computer word is a simple and natural way
to compress the representation of a string and to speed up its processing.
Exploiting this idea, we propose an index for a packed string, based on a {\em
sparse suffix tree} \cite{KU-96} with appropriately defined suffix links.
Assuming, under the standard unit-cost RAM model, that a word can store up to
characters ( the alphabet size), our index takes
space, i.e. the same space as the packed string itself.
The resulting pattern matching algorithm runs in time ,
where is the length of the pattern, is the actual number of characters
stored in a word and is the number of pattern occurrences
On the sub-permutations of pattern avoiding permutations
There is a deep connection between permutations and trees. Certain
sub-structures of permutations, called sub-permutations, bijectively map to
sub-trees of binary increasing trees. This opens a powerful tool set to study
enumerative and probabilistic properties of sub-permutations and to investigate
the relationships between 'local' and 'global' features using the concept of
pattern avoidance. First, given a pattern {\mu}, we study how the avoidance of
{\mu} in a permutation {\pi} affects the presence of other patterns in the
sub-permutations of {\pi}. More precisely, considering patterns of length 3, we
solve instances of the following problem: given a class of permutations K and a
pattern {\mu}, we ask for the number of permutations whose
sub-permutations in K satisfy certain additional constraints on their size.
Second, we study the probability for a generic pattern to be contained in a
random permutation {\pi} of size n without being present in the
sub-permutations of {\pi} generated by the entry . These
theoretical results can be useful to define efficient randomized pattern-search
procedures based on classical algorithms of pattern-recognition, while the
general problem of pattern-search is NP-complete
Order preserving pattern matching on trees and DAGs
The order preserving pattern matching (OPPM) problem is, given a pattern
string and a text string , find all substrings of which have the
same relative orders as . In this paper, we consider two variants of the
OPPM problem where a set of text strings is given as a tree or a DAG. We show
that the OPPM problem for a single pattern of length and a text tree
of size can be solved in time if the characters of are
drawn from an integer alphabet of polynomial size. The time complexity becomes
if the pattern is over a general ordered alphabet. We
then show that the OPPM problem for a single pattern and a text DAG is
NP-complete
Pattern-based phylogenetic distance estimation and tree reconstruction
We have developed an alignment-free method that calculates phylogenetic
distances using a maximum likelihood approach for a model of sequence change on
patterns that are discovered in unaligned sequences. To evaluate the
phylogenetic accuracy of our method, and to conduct a comprehensive comparison
of existing alignment-free methods (freely available as Python package decaf+py
at http://www.bioinformatics.org.au), we have created a dataset of reference
trees covering a wide range of phylogenetic distances. Amino acid sequences
were evolved along the trees and input to the tested methods; from their
calculated distances we infered trees whose topologies we compared to the
reference trees.
We find our pattern-based method statistically superior to all other tested
alignment-free methods on this dataset. We also demonstrate the general
advantage of alignment-free methods over an approach based on automated
alignments when sequences violate the assumption of collinearity. Similarly, we
compare methods on empirical data from an existing alignment benchmark set that
we used to derive reference distances and trees. Our pattern-based approach
yields distances that show a linear relationship to reference distances over a
substantially longer range than other alignment-free methods. The pattern-based
approach outperforms alignment-free methods and its phylogenetic accuracy is
statistically indistinguishable from alignment-based distances.Comment: 21 pages, 3 figures, 2 table
Generating trees and pattern avoidance in alternating permutations
We extend earlier work of the same author to enumerate alternating
permutations avoiding the permutation pattern 2143. We use a generating tree
approach to construct a recursive bijection between the set A_{2n}(2143) of
alternating permutations of length 2n avoiding 2143 and standard Young tableaux
of shape (n, n, n) and between the set A_{2n + 1}(2143) of alternating
permutations of length 2n + 1 avoiding 2143 and shifted standard Young tableaux
of shape (n + 2, n + 1, n). We also give a number of conjectures and open
questions on pattern avoidance in alternating permutations and generalizations
thereof.Comment: 21 pages. To be presented at FPSAC 2010. Comments welcome
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