531 research outputs found
Enumerating five families of pattern-avoiding inversion sequences; and introducing the powered Catalan numbers
The first problem addressed by this article is the enumeration of some
families of pattern-avoiding inversion sequences. We solve some enumerative
conjectures left open by the foundational work on the topics by Corteel et al.,
some of these being also solved independently by Lin, and Kim and Lin. The
strength of our approach is its robustness: we enumerate four families of pattern-avoiding inversion sequences
ordered by inclusion using the same approach. More precisely, we provide a
generating tree (with associated succession rule) for each family which
generalizes the one for the family .
The second topic of the paper is the enumeration of a fifth family of
pattern-avoiding inversion sequences (containing ). This enumeration is
also solved \emph{via} a succession rule, which however does not generalize the
one for . The associated enumeration sequence, which we call the
\emph{powered Catalan numbers}, is quite intriguing, and further investigated.
We provide two different succession rules for it, denoted and
, and show that they define two types of families enumerated
by powered Catalan numbers. Among such families, we introduce the \emph{steady
paths}, which are naturally associated with . They allow us to
bridge the gap between the two types of families enumerated by powered Catalan
numbers: indeed, we provide a size-preserving bijection between steady paths
and valley-marked Dyck paths (which are naturally associated with
).
Along the way, we provide several nice connections to families of
permutations defined by the avoidance of vincular patterns, and some
enumerative conjectures.Comment: V2 includes modifications suggested by referees (in particular, a
much shorter Section 3, to account for arXiv:1706.07213
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
Grand Dyck paths with air pockets
Grand Dyck paths with air pockets (GDAP) are a generalization of Dyck paths
with air pockets by allowing them to go below the -axis. We present
enumerative results on GDAP (or their prefixes) subject to various restrictions
such as maximal/minimal height, ordinate of the last point and particular first
return decomposition. In some special cases we give bijections with other known
combinatorial classes.Comment: 20 pages, 4 figure
IST Austria Thesis
This dissertation focuses on algorithmic aspects of program verification, and presents modeling and complexity advances on several problems related to the
static analysis of programs, the stateless model checking of concurrent programs, and the competitive analysis of real-time scheduling algorithms.
Our contributions can be broadly grouped into five categories.
Our first contribution is a set of new algorithms and data structures for the quantitative and data-flow analysis of programs, based on the graph-theoretic notion of treewidth.
It has been observed that the control-flow graphs of typical programs have special structure, and are characterized as graphs of small treewidth.
We utilize this structural property to provide faster algorithms for the quantitative and data-flow analysis of recursive and concurrent programs.
In most cases we make an algebraic treatment of the considered problem,
where several interesting analyses, such as the reachability, shortest path, and certain kind of data-flow analysis problems follow as special cases.
We exploit the constant-treewidth property to obtain algorithmic improvements for on-demand versions of the problems,
and provide data structures with various tradeoffs between the resources spent in the preprocessing and querying phase.
We also improve on the algorithmic complexity of quantitative problems outside the algebraic path framework,
namely of the minimum mean-payoff, minimum ratio, and minimum initial credit for energy problems.
Our second contribution is a set of algorithms for Dyck reachability with applications to data-dependence analysis and alias analysis.
In particular, we develop an optimal algorithm for Dyck reachability on bidirected graphs, which are ubiquitous in context-insensitive, field-sensitive points-to analysis.
Additionally, we develop an efficient algorithm for context-sensitive data-dependence analysis via Dyck reachability,
where the task is to obtain analysis summaries of library code in the presence of callbacks.
Our algorithm preprocesses libraries in almost linear time, after which the contribution of the library in the complexity of the client analysis is (i)~linear in the number of call sites and (ii)~only logarithmic in the size of the whole library, as opposed to linear in the size of the whole library.
Finally, we prove that Dyck reachability is Boolean Matrix Multiplication-hard in general, and the hardness also holds for graphs of constant treewidth.
This hardness result strongly indicates that there exist no combinatorial algorithms for Dyck reachability with truly subcubic complexity.
Our third contribution is the formalization and algorithmic treatment of the Quantitative Interprocedural Analysis framework.
In this framework, the transitions of a recursive program are annotated as good, bad or neutral, and receive a weight which measures
the magnitude of their respective effect.
The Quantitative Interprocedural Analysis problem asks to determine whether there exists an infinite run of the program where the long-run ratio of the bad weights over the good weights is above a given threshold.
We illustrate how several quantitative problems related to static analysis of recursive programs can be instantiated in this framework,
and present some case studies to this direction.
Our fourth contribution is a new dynamic partial-order reduction for the stateless model checking of concurrent programs. Traditional approaches rely on the standard Mazurkiewicz equivalence between traces, by means of partitioning the trace space into equivalence classes, and attempting to explore a few representatives from each class.
We present a new dynamic partial-order reduction method called the Data-centric Partial Order Reduction (DC-DPOR).
Our algorithm is based on a new equivalence between traces, called the observation equivalence.
DC-DPOR explores a coarser partitioning of the trace space than any exploration method based on the standard Mazurkiewicz equivalence.
Depending on the program, the new partitioning can be even exponentially coarser.
Additionally, DC-DPOR spends only polynomial time in each explored class.
Our fifth contribution is the use of automata and game-theoretic verification techniques in the competitive analysis and synthesis of real-time scheduling algorithms for firm-deadline tasks.
On the analysis side, we leverage automata on infinite words to compute the competitive ratio of real-time schedulers subject to various environmental constraints.
On the synthesis side, we introduce a new instance of two-player mean-payoff partial-information games, and show
how the synthesis of an optimal real-time scheduler can be reduced to computing winning strategies in this new type of games
The parallel TASEP, fixed particle number and weighted Motzkin paths
In this paper the totally asymmetric exclusion process (TASEP) with parallel
update on an open lattice of size is considered in the maximum-current
region. A formal expression for the generating function for the weight of
configurations with particles is given. Further an interpretation in terms
of -colored weighted Motzkin paths is presented. Using previous
results (Woelki and Schreckenberg 2009 {\it J. Stat. Mech} P05014, Woelki 2010
{\it Cellular Automata}, pp 637-645) the generating function is compared with
the one for a possible 2nd-class particle dynamics for the parallel TASEP. It
is shown that both become physically equivalent in the thermodynamic limit.Comment: 11 pages, 3 figure
Combinatorial and analytic techniques for lattice path enumeration
This bachelor s degree thesis studies two type of combinatorial objects. The first ones are exact models of which we can get exact counting formulas, whereas the second type of models are the ones of which we don t have enumerative exact results. The first part of the work focuses on the exact models and two methods useful to find counting formulas are described: the Symbolic method and the Kernel method. In the second part of this thesis, an specific type of non exact model is addressed: self-avoiding walks. Although there are no exact counting formulas, we are going to study its asymptotic behaviour and we will proof a theorem which states that the connective constant of self-avoiding walks in the hexagonal lattice H equals √2 + √2
- …