Random lattice triangulations: Structure and algorithms

The paper concerns lattice triangulations, that is, triangulations of the integer points in a polygon in $\mathbb{R}^2$ whose vertices are also integer points. Lattice triangulations have been studied extensively both as geometric objects in their own right and by virtue of applications in algebraic geometry. Our focus is on random triangulations in which a triangulation $\sigma$ has weight $\lambda^{|\sigma|}$, where $\lambda$ is a positive real parameter, and $|\sigma|$ is the total length of the edges in $\sigma$. Empirically, this model exhibits a "phase transition" at $\lambda=1$ (corresponding to the uniform distribution): for $\lambda<1$ distant edges behave essentially independently, while for $\lambda>1$ very large regions of aligned edges appear. We substantiate this picture as follows. For $\lambda<1$ sufficiently small, we show that correlations between edges decay exponentially with distance (suitably defined), and also that the Glauber dynamics (a local Markov chain based on flipping edges) is rapidly mixing (in time polynomial in the number of edges in the triangulation). This dynamics has been proposed by several authors as an algorithm for generating random triangulations. By contrast, for $\lambda>1$ we show that the mixing time is exponential. These are apparently the first rigorous quantitative results on the structure and dynamics of random lattice triangulations.Comment: Published at http://dx.doi.org/10.1214/14-AAP1033 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

The Flip Diameter of Rectangulations and Convex Subdivisions

We study the configuration space of rectangulations and convex subdivisions of $n$ points in the plane. It is shown that a sequence of $O(n\log n)$ elementary flip and rotate operations can transform any rectangulation to any other rectangulation on the same set of $n$ points. This bound is the best possible for some point sets, while $\Theta(n)$ operations are sufficient and necessary for others. Some of our bounds generalize to convex subdivisions of $n$ points in the plane.Comment: 17 pages, 12 figures, an extended abstract has been presented at LATIN 201

On the number of higher order Delaunay triangulations

"Vegeu el resum a l'inici del document del fitxer adjunt"

On the number of drawings of a combinatorial triangulations

Aquesta tesi explora la relació entre triangulacions combinatòries i geomètriques en geometria discreta i combinatòria. Per triangulacions combinatòries ens referim a grafs, mentre que per triangulacions geomètriques ens referim a dibuixos de grafs com a plans maximals amb línies rectes com a arestes sobre un conjunt de punts fixat al pla. Estudiem de quantes maneres es pot traçar una triangulació combinatòria com a triangulació geomètrica sobre un conjunt de punts donat. La nostra contribució central és demostrar que una triangulació combinatòria fixa amb n vèrtexs es pot dibuixar d'almenys 1,31^n maneres en un conjunt de n punts com a diferents triangulacions geomètriques. També analitzem els límits superiors i una versió acolorida daquest problema. L'enfocament suggerit pot ajudar a avançar en la resolució del problema obert per limitar el nombre de triangulacions geomètriques. A més, aprofundim en fonaments històrics, com el treball de Tutte, que proporciona el nombre exacte de triangulacions combinatòries amb n vèrtexs.Esta tesis explora la relación entre triangulaciones combinatorias y geométricas en geometría discreta y combinatoria. Con triangulaciones combinatorias nos referimos a grafos, mientras que con triangulaciones geométricas nos referimos a dibujos de grafos como planos maximales con líneas rectas como aristas sobre un conjunto de puntos fijado en el plano. Estudiamos de cuántas maneras se puede trazar una triangulación combinatoria como triangulación geométrica sobre un conjunto de puntos dado. Nuestra contribución central es demostrar que una triangulación combinatoria fija con n vértices se puede dibujar de al menos 1,31^n maneras en un conjunto de n puntos como diferentes triangulaciones geométricas. También analizamos los límites superiores y una versión coloreada de este problema. El enfoque sugerido puede ayudar a avanzar en la resolución del problema abierto para limitar el número de triangulaciones geométricas. Además, profundizamos en fundamentos históricos, como el trabajo de Tutte, que proporciona el número exacto de triangulaciones combinatorias con n vértices.This thesis explores the intricate relationship between combinatorial and geometric triangulations in discrete and combinatorial geometry. With combinatorial triangulations we refer to graphs, while with geometric triangulations we refer to maximal planar straight-line drawings on a point set in the plane. We study in how many ways can a combinatorial triangulation be drawn as geometric triangulation on a given point set. Our central contribution is proving that a fixed combinatorial triangulation with n vertices can be drawn in at least 1.31^n ways in a set of n points as different geometric triangulations. We also discuss upper bounds and a colored version of this problem. The suggested approach may help to advance the resolution of the open problem to bound the number of geometric triangulations. Additionally, we delve into historical foundations, such as Tutte's work, which provides the exact number of combinatorial triangulations with n vertices

Algorithmic Graph Theory

The main focus of this workshop was on mathematical techniques needed for the development of efficient solutions and algorithms for computationally difficult graph problems. The techniques studied at the workshhop included: the probabilistic method and randomized algorithms, approximation and optimization, structured families of graphs and approximation algorithms for large problems. The workshop Algorithmic Graph Theory was attended by 46 participants, many of them being young researchers. In 15 survey talks an overview of recent developments in Algorithmic Graph Theory was given. These talks were supplemented by 10 shorter talks and by two special sessions

TRIANGULATION PROBLEMS ON GEOMETRIC GRAPHS - SAMPLING OVER CONVEX TRIANGULATIONS

Γεωμετρικό γράφημα καλείται ένα σύνολο σημείων V στο επίπεδο μαζί με ένα σύνολο ευθυγράμμων τμημάτων (ακμών) E που έχουν τα άκρα τους στο V, και εύκολα συσχετίζεται με το &quot;αφηρημένο&quot; γράφημα G(V,E). Μελετώντας το πάχος του, δηλαδή τη διαμέριση των ακμών του σε υποσύνολα ελεύθερα διασταυρώσεων (ένα NP-δύσκολο πρόβλημα βελτιστοποίησης), προκύπτει και το πρόβλημα της ύπαρξης τριγωνοποίησης ως ένα ελεύθερο διασταυρώσεων υποσύνολο T των ακμών, καθώς μια τριγωνοποίηση του V αποτελεί το μέγιστο δυνατό τέτοιο σύνολο που είναι δυνατόν να οριστεί δεδομένου του V. Η Διπλωματική αυτή Εργασία αφορά στη μελέτη μιας οικογένειας προβλημάτων ύπαρξης τριγωνοποίησης και την ταξινόμησή τους ως προς την πολυπλοκότητα απόφασης, αλλά και μέτρησης. Από αυτά, το γενικό πρόβλημα απόφασης είναι το μόνο μελετημένο στη βιβλιογραφία (Lloyd, 1977, NP-δύσκολο), ενώ εμείς μελετάμε αφενός την ειδική περίπτωση των κυρτών γεωμετρικών γραφημάτων, αφετέρου ένα &quot;ενδιάμεσο&quot; πρόβλημα ύπαρξης τριγωνοποιημένου πολυγώνου, δημιουργώντας έναν νέο 2 x 2 πίνακα αποτελεσμάτων. Στο τελευταίο κεφάλαιο, τροποποιούμε το πλαίσιο της δουλειάς μας έτσι ώστε να κατασκευάσουμε έναν αλγόριθμο για ομοιόμορφη δειγματοληψία και βέλτιστη κωδικοποίηση των κυρτών τριγωνοποιήσεων, ο οποίος υπερέχει έναντι κάθε γνωστού αλγορίθμου έως σήμερα.A geometric graph is a set of points V on the plane and a set of straight line segments E with endpoints in V, potentially and instinctively associated with the abstract G(V,E). When studying its thickness, i.e. partitioning its edges into crossing-free subsets (an NP-hard optimization problem), the problem of triangulation existence as a crossing-free subset T of the edges naturally occurs, as a triangulation of V is the largest such possible set that may be defined on V. In this Thesis, we examine a family of triangulation existence problems and classify them with respect to their complexity, both for their decision and their counting versions. The general case decision problem is the only one appearing in bibliography (Lloyd, 1977, NP-hard), while we deal with the convex case restriction and an &quot;intermediate&quot; polygon triangulation existence problem, fixing a new 2 by 2 table of results. In the final chapter, we modify our framework in order to build an exact uniform sampling and optimal coding algorithm for convex triangulations, which outperforms any known algorithm to date

Compression sans perte de maillages triangulaires adaptée aux applications métrologiques

La compression est un incontournable lorsque des modèles triangulaires 3D massifs doivent être transmis via un réseau de communication. La compression se doit d'être sans perte lorsque les modèles sont utilisés à des fins métrologiques. Cependant, les modèles capturés par scanneurs 3D contiennent généralement des artefacts de numérisation tels que la présence de trous dans le maillage, de petits regroupements distincts de triangles sous forme de surfaces ou de volumes ainsi que de singularités non-manifold (c.-à-d. un sommet appartenant à deux regroupement de triangles distincts). Ces aberrations rendent les techniques de compression standards inaptes à compresser sans échec le modèle. Ce mémoire propose une extension à une technique de compression et décompression sans perte des données topologiques nommée Edgebreaker. Le remplissage des trous par l'addition d'un sommet, l'insertion de faces triangulaires afin de lier les îlots ainsi que la duplication des sommets non-manifold sont proposées comme étapes de prétraitement afin de rendre le modèle compatible avec l'approche standard d'Edgebreaker. Les résultats obtenus démontrent que la solution proposée permet la compression sans perte de modèles hautement bruités à de hauts taux de compression. Les taux de compression résultants obtenus par notre approche se comparent également avec les taux observables pour des modèles sans imperfections compressés par Edgebreaker

The effects of bias on sampling algorithms and combinatorial objects

Markov chains are algorithms that can provide critical information from exponentially large sets efficiently through random sampling. These algorithms are ubiquitous across numerous scientific and engineering disciplines, including statistical physics, biology and operations research. In this thesis we solve sampling problems at the interface of theoretical computer science with applied computer science, discrete mathematics, statistical physics, chemistry and economics. A common theme throughout each of these problems is the use of bias. The first problem we study is biased permutations which arise in the context of self-organizing lists. Here we are interested in the mixing time of a Markov chain that performs nearest neighbor transpositions in the non-uniform setting. We are given "positively biased'' probabilities $\{p_{i,j} \geq 1/2 \}$ for all $i < j$ and let $p_{j,i} = 1-p_{i,j}$. In each step, the chain chooses two adjacent elements~$k,$ and~$\ell$ and exchanges their positions with probability $p_{ \ell, k}$. We define two general classes of bias and give the first proofs that the chain is rapidly mixing for both. We also demonstrate that the chain is not always rapidly mixing by constructing an example requiring exponential time to converge to equilibrium. Next we study rectangular dissections of an $n \times n$ lattice region into rectangles of area $n$, where $n=2^k$ for an even integer $k.$ We consider a weighted version of a natural edge flipping Markov chain where, given a parameter $\lambda > 0,$ we would like to generate each rectangular dissection (or dyadic tiling)~$\sigma$ with probability proportional to $\lambda^{|\sigma|},$ where $|\sigma|$ is the total edge length. First we look at the restricted case of dyadic tilings, where each rectangle is required to have the form $R = [s2^{u},(s+1)2^{u}]\times [t2^{v},(t+1)2^{v}],$ where $s, t, u$ and~$v$ are nonnegative integers. Here we show there is a phase transition: when $\lambda 1,$ the mixing time is $\exp(\Omega({n^2}))$. The behavior for general rectangular dissections is more subtle, and we show the chain requires exponential time when $\lambda >1$ and when $\lambda <1.$ The last two problems we study arise directly from applications in chemistry and economics. Colloids are binary mixtures of molecules with one type of molecule suspended in another. It is believed that at low density typical configurations will be well-mixed throughout, while at high density they will separate into clusters. We characterize the high and low density phases for a general family of discrete interfering colloid models by showing that they exhibit a "clustering property" at high density and not at low density. The clustering property states that there will be a region that has very high area to perimeter ratio and very high density of one type of molecule. A special case is mixtures of squares and diamonds on $\Z^2$ which correspond to the Ising model at fixed magnetization. Subsequently, we expanded techniques developed in the context of colloids to give a new rigorous underpinning to the Schelling model, which was proposed in 1971 by economist Thomas Schelling to understand the causes of racial segregation. Schelling considered residents of two types, where everyone prefers that the majority of his or her neighbors are of the same type. He showed through simulations that even mild preferences of this type can lead to segregation if residents move whenever they are not happy with their local environments. We generalize the Schelling model to include a broad class of bias functions determining individuals happiness or desire to move. We show that for any influence function in this class, the dynamics will be rapidly mixing and cities will be integrated if the racial bias is sufficiently low. However when the bias is sufficiently high, we show the dynamics take exponential time to mix and a large cluster of one type will form.Ph.D