Analytic Combinatorics of Lattice Paths with Forbidden Patterns, the Vectorial Kernel Method, and Generating Functions for Pushdown Automata

Algorithmica ◽  
2019 ◽  
Vol 82 (3) ◽  
pp. 386-428 ◽  
Author(s):  
Andrei Asinowski ◽  
Axel Bacher ◽  
Cyril Banderier ◽  
Bernhard Gittenberger
Keyword(s):  
Kernel Method ◽  
Lattice Paths ◽  
Integer Sequences ◽  
Unified Framework ◽  
Analytic Combinatorics ◽  
On Line ◽  
Forbidden Patterns ◽  
Motzkin Paths ◽  
Self Avoiding Walks ◽  
Pushdown Automata

Abstract In this article we develop a vectorial kernel method—a powerful method which solves in a unified framework all the problems related to the enumeration of words generated by a pushdown automaton. We apply it for the enumeration of lattice paths that avoid a fixed word (a pattern), or for counting the occurrences of a given pattern. We unify results from numerous articles concerning patterns like peaks, valleys, humps, etc., in Dyck and Motzkin paths. This refines the study by Banderier and Flajolet from 2002 on enumeration and asymptotics of lattice paths: we extend here their results to pattern-avoiding walks/bridges/meanders/excursions. We show that the autocorrelation polynomial of this forbidden pattern, as introduced by Guibas and Odlyzko in 1981 in the context of rational languages, still plays a crucial role for our algebraic languages. En passant, our results give the enumeration of some classes of self-avoiding walks, and prove several conjectures from the On-Line Encyclopedia of Integer Sequences. Finally, we also give the trivariate generating function (length, final altitude, number of occurrences of the pattern p), and we prove that the number of occurrences is normally distributed and linear with respect to the length of the walk: this is what Flajolet and Sedgewick call an instance of Borges’s theorem.

scholarly journals Analytic Combinatorics of Lattice Paths: Enumeration and Asymptotics for the Area

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Cyril Banderier ◽  
Bernhard Gittenberger
Keyword(s):  
Symmetric Functions ◽  
Kernel Method ◽  
Brownian Bridge ◽  
Lattice Paths ◽  
Reflected Brownian Motion ◽  
Analytic Combinatorics ◽  
Average Area ◽  
First Case ◽  
Finite Set ◽  
International Audience

International audience This paper tackles the enumeration and asymptotics of the area below directed lattice paths (walks on $\mathbb{N}$ with a finite set of jumps). It is a nice surprise (obtained via the "kernel method'') that the generating functions of the moments of the area are algebraic functions, expressible as symmetric functions in terms of the roots of the kernel. For a large class of walks, we give full asymptotics for the average area of excursions ("discrete'' reflected Brownian bridge) and meanders ("discrete'' reflected Brownian motion). We show that drift is not playing any role in the first case. We also generalise previous works related to the number of points below a path and to the area between a path and a line of rational slope.

scholarly journals Kinematic and dynamic model-based control of wheeled mobile manipulators: a unified framework for reactive approaches

Robotica ◽  
2007 ◽  
Vol 25 (2) ◽  
pp. 157-173 ◽  
Author(s):  
V. Padois ◽  
J.-Y. Fourquet ◽  
P. Chiron
Keyword(s):  
Dynamic Models ◽  
Mobile Manipulation ◽  
Mobile Manipulators ◽  
Reactive Control ◽  
Unified Framework ◽  
Modelling Framework ◽  
On Line ◽  
Real Robot ◽  
Wheeled Mobile Manipulators ◽  
Line Switching

SUMMARYThe work presented in this paper aims at providing a unified modelling framework for the reactive control of wheeled mobile manipulators (WMM). Where most work in the literature often provides models, sometimes simplified, of a given type of WMM, an extensive description of obtaining explicit kinematic and dynamic models of those systems is given. This modelling framework is particularly well suited for reactive control approaches, which, in the case of mobile manipulation missions, are often necessary to handle the complexity of the tasks to be fulfilled, the dynamic aspect of the extended workspace and the uncertainties on the knowledge of the environment. A flexible reactive framework is thus also provided, allowing the sequencing of operational tasks (in our case, tasks described in the end-effector frame) whose natures are different but also an on-line switching mechanism between constraints that are to be satisfied using the system redundancy. This framework has been successfully implemented in simulation and on a real robot. Some of the obtained results are presented.

scholarly journals Slicings of Parallelogram Polyominoes: Catalan, Schröder, Baxter, and Other Sequences

10.37236/7375 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Nicholas R. Beaton ◽  
Mathilde Bouvel ◽  
Veronica Guerrini ◽  
Simone Rinaldi
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Kernel Method ◽  
Lattice Paths ◽  
Schröder Numbers ◽  
Special Cases ◽  
Generating Tree

We provide a new succession rule (i.e. generating tree) associated with Schröder numbers, that interpolates between the known succession rules for Catalan and Baxter numbers. We define Schröder and Baxter generalizations of parallelogram polyominoes, called slicings, which grow according to these succession rules. In passing, we also exhibit Schröder subclasses of Baxter classes, namely a Schröder subset of triples of non-intersecting lattice paths, a new Schröder subset of Baxter permutations, and a new Schröder subset of mosaic floorplans. Finally, we define two families of subclasses of Baxter slicings: the $m$-skinny slicings and the $m$-row-restricted slicings, for $m \in \mathbb{N}$. Using functional equations and the kernel method, their generating functions are computed in some special cases, and we conjecture that they are algebraic for any $m$.

On-Line Defect Detecting Method Based on Kernel Method

2011 ◽  
Vol 474-476 ◽  
pp. 858-863 ◽  
Author(s):  
Ke Jia Xu ◽  
Bin Chen ◽  
Li Zeng
Keyword(s):  
Real Time ◽  
Kernel Method ◽  
Principal Component ◽  
Kernel Principal Component Analysis ◽  
Image Method ◽  
Speed Up ◽  
On Line ◽  
And Performance ◽  
Detecting Method ◽  
Better Than

The conflict between accuracy and speed is one of the most well-known dilemmas of the real-time defect detecting system. This paper presents a real-time defect detecting algorithm based on Kernel principal component analysis (KPCA). KPCA-based feature extraction have recently shown to be very effective for image denoising, however the Normal KPCA method is time-consuming. In our method, we propose a progressive algorithm to speed up the reconstruct process while improve accuracy. Experimental results demonstrate that our method is dramatically better than Normal KPCA Pre-image method in terms of speed and performance.

scholarly journals Asymptotics of lattice walks via analytic combinatorics in several variables

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Stephen Melczer ◽  
Mark C. Wilson
Keyword(s):  
Generating Functions ◽  
Kernel Method ◽  
Rational Functions ◽  
Two Dimensional ◽  
Combinatorial Properties ◽  
Analytic Combinatorics ◽  
Several Variables ◽  
Algebra Approach ◽  
Finite Set ◽  
International Audience

International audience We consider the enumeration of walks on the two-dimensional non-negative integer lattice with steps defined by a finite set S ⊆ {±1, 0}2 . Up to isomorphism there are 79 unique two-dimensional models to consider, and previous work in this area has used the kernel method, along with a rigorous computer algebra approach, to show that 23 of the 79 models admit D-finite generating functions. In 2009, Bostan and Kauers used Pade ́-Hermite approximants to guess differential equations which these 23 generating functions satisfy, in the process guessing asymptotics of their coefficient sequences. In this article we provide, for the first time, a complete rigorous verification of these guesses. Our technique is to use the kernel method to express 19 of the 23 generating functions as diagonals of tri-variate rational functions and apply the methods of analytic combinatorics in several variables (the remaining 4 models have algebraic generating functions and can thus be handled by univariate techniques). This approach also shows the link between combinatorial properties of the models and features of its asymptotics such as asymptotic and polynomial growth factors. In addition, we give expressions for the number of walks returning to the x-axis, the y-axis, and the origin, proving recently conjectured asymptotics of Bostan, Chyzak, van Hoeij, Kauers, and Pech.

scholarly journals Self-weighted Multiple Kernel Learning for Graph-based Clustering and Semi-supervised Classification

2018 ◽  
Author(s):  
Zhao Kang ◽  
Xiao Lu ◽  
Jinfeng Yi ◽  
Zenglin Xu
Keyword(s):  
Supervised Classification ◽  
Kernel Method ◽  
Empirical Studies ◽  
Multiple Kernel Learning ◽  
Kernel Learning ◽  
Unified Framework ◽  
Kernel Weights ◽  
Multiple Kernel ◽  
Benchmark Datasets ◽  
Graph Based Clustering

Multiple kernel learning (MKL) method is generally believed to perform better than single kernel method. However, some empirical studies show that this is not always true: the combination of multiple kernels may even yield an even worse performance than using a single kernel. There are two possible reasons for the failure: (i) most existing MKL methods assume that the optimal kernel is a linear combination of base kernels, which may not hold true; and (ii) some kernel weights are inappropriately assigned due to noises and carelessly designed algorithms. In this paper, we propose a novel MKL framework by following two intuitive assumptions: (i) each kernel is a perturbation of the consensus kernel; and (ii) the kernel that is close to the consensus kernel should be assigned a large weight. Impressively, the proposed method can automatically assign an appropriate weight to each kernel without introducing additional parameters, as existing methods do. The proposed framework is integrated into a unified framework for graph-based clustering and semi-supervised classification. We have conducted experiments on multiple benchmark datasets and our empirical results verify the superiority of the proposed framework.

scholarly journals Vertically Constrained Motzkin-Like Paths Inspired by Bobbin Lace

10.37236/7799 ◽  
2019 ◽  
Vol 26 (2) ◽  
Author(s):  
Veronika Irvine ◽  
Stephen Melczer ◽  
Frank Ruskey
Keyword(s):  
Mathematical Model ◽  
Generating Functions ◽  
Finite Lattice ◽  
Lattice Paths ◽  
Change Direction ◽  
Quarter Plane ◽  
Directed Paths ◽  
Schröder Paths ◽  
Motzkin Paths

Inspired by a new mathematical model for bobbin lace, this paper considers finite lattice paths formed from the set of step vectors $\mathfrak{A}=$$\{\rightarrow,$ $\nearrow,$ $\searrow,$ $\uparrow,$ $\downarrow\}$ with the restriction that vertical steps $(\uparrow, \downarrow)$ cannot be consecutive. The set $\mathfrak{A}$ is the union of the well known Motzkin step vectors $\mathfrak{M}=$$\{\rightarrow,$ $\nearrow,$ $\searrow\}$ with the vertical steps $\{\uparrow, \downarrow\}$. An explicit bijection $\phi$ between the exhaustive set of vertically constrained paths formed from $\mathfrak{A}$ and a bisection of the paths generated by $\mathfrak{M}S$ is presented. In a similar manner, paths with the step vectors $\mathfrak{B}=$$\{\nearrow,$ $\searrow,$ $\uparrow,$ $\downarrow\}$, the union of Dyck step vectors and constrained vertical steps, are examined.  We show, using the same $\phi$ mapping, that there is a bijection between vertically constrained $\mathfrak{B}$ paths and the subset of Motzkin paths avoiding horizontal steps at even indices.  Generating functions are derived to enumerate these vertically constrained, partially directed paths when restricted to the half and quarter-plane.  Finally, we extend Schröder and Delannoy step sets in a similar manner and find a bijection between these paths and a subset of Schröder paths that are smooth (do not change direction) at a regular horizontal interval.

scholarly journals Basic analytic combinatorics of directed lattice paths

2002 ◽  
Vol 281 (1-2) ◽  
pp. 37-80 ◽  
Author(s):  
Cyril Banderier ◽  
Philippe Flajolet
Keyword(s):  
Lattice Paths ◽  
Analytic Combinatorics
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