Catalan's intervals and realizers of triangulations

The Stanley lattice, Tamari lattice and Kreweras lattice are three remarkable orders defined on the set of Catalan objects of a given size. These lattices are ordered by inclusion: the Stanley lattice is an extension of the Tamari lattice which is an extension of the Kreweras lattice. The Stanley order can be defined on the set of Dyck paths of size $n$ as the relation of \emph{being above}. Hence, intervals in the Stanley lattice are pairs of non-crossing Dyck paths. In a former article, the second author defined a bijection $\Phi$ between pairs of non-crossing Dyck paths and the realizers of triangulations (or Schnyder woods). We give a simpler description of the bijection $\Phi$. Then, we study the restriction of $\Phi$ to Tamari's and Kreweras' intervals. We prove that $\Phi$ induces a bijection between Tamari intervals and minimal realizers. This gives a bijection between Tamari intervals and triangulations. We also prove that $\Phi$ induces a bijection between Kreweras intervals and the (unique) realizers of stack triangulations. Thus, $\Phi$ induces a bijection between Kreweras intervals and stack triangulations which are known to be in bijection with ternary trees.Comment: 22 page

Asymptotic of geometrical navigation on a random set of points of the plane

A navigation on a set of points $S$ is a rule for choosing which point to move to from the present point in order to progress toward a specified target. We study some navigations in the plane where $S$ is a non uniform Poisson point process (in a finite domain) with intensity going to $+\infty$. We show the convergence of the traveller path lengths, the number of stages done, and the geometry of the traveller trajectories, uniformly for all starting points and targets, for several navigations of geometric nature. Other costs are also considered. This leads to asymptotic results on the stretch factors of random Yao-graphs and random $\theta$-graphs

The Stretch Factor of L1L_1- and L∞L_\infty-Delaunay Triangulations

In this paper we determine the stretch factor of the $L_1$-Delaunay and $L_\infty$-Delaunay triangulations, and we show that this stretch is $\sqrt{4+2\sqrt{2}} \approx 2.61$. Between any two points $x,y$ of such triangulations, we construct a path whose length is no more than $\sqrt{4+2\sqrt{2}}$ times the Euclidean distance between $x$ and $y$, and this bound is best possible. This definitively improves the 25-year old bound of $\sqrt{10}$ by Chew (SoCG '86). To the best of our knowledge, this is the first time the stretch factor of the well-studied $L_p$-Delaunay triangulations, for any real $p\ge 1$, is determined exactly

There are Plane Spanners of Maximum Degree 4

Let E be the complete Euclidean graph on a set of points embedded in the plane. Given a constant t >= 1, a spanning subgraph G of E is said to be a t-spanner, or simply a spanner, if for any pair of vertices u,v in E the distance between u and v in G is at most t times their distance in E. A spanner is plane if its edges do not cross. This paper considers the question: "What is the smallest maximum degree that can always be achieved for a plane spanner of E?" Without the planarity constraint, it is known that the answer is 3 which is thus the best known lower bound on the degree of any plane spanner. With the planarity requirement, the best known upper bound on the maximum degree is 6, the last in a long sequence of results improving the upper bound. In this paper we show that the complete Euclidean graph always contains a plane spanner of maximum degree at most 4 and make a big step toward closing the question. Our construction leads to an efficient algorithm for obtaining the spanner from Chew's L1-Delaunay triangulation

Aspects algorithmiques et combinatoires des réaliseurs des graphes plans maximaux

Les réaliseurs, ou arbres de Schnyder, ont été introduits par Walter Schnyder à la fin des années 80 pour caractériser les graphes planaires, puis pour dessiner ces mêmes graphes sur des grilles (n-2)x(n-2). Dans ce document nous proposons dans un premier temps une extension du théorème de Wagner aux réaliseurs, qui nous permet d'établir une relation entre le nombre de feuilles et le nombre de faces tricolores d'un réaliseur. Ensuite, à l'aide d'une bijection entre les réaliseurs et les paires de chemins de Dyck qui ne se coupent pas, nous énumérons les réaliseurs. Un algorithme de génération aléatoire de p chemins de Dyck ne se coupant pas, est également présenté. Il permet en outre de générer aléatoirement des réaliseurs en temps linéaire. Puis nous montrons que grâce aux réaliseurs, il est possible de dessiner, à l'aide de lignes brisées des graphes planaires sur des grilles de largeur et de surface optimales. Enfin, nous proposons une généralisation des réaliseurs minimaux aux graphes planaires connexes : les arbres recouvrants bien-ordonnés. Grâce à cette généralisation ainsi qu'à une méthode de triangulation adaptée nous proposons un algorithme de codage des graphes planaires à n sommets en 5,007n bits.The realizers, or Schnyder trees, have introduced by Walter Schnyder in the late 80's to give a characterization of planar graphs and to draw them on (n-2)x(n-2) grids. In this document, we first give an extension of Wagner's theorem to realizers. Using this theorem we establish a relationship between the number of leaves and the number of 3-colored faces of a realizer. A bijection between realizers and pairs of non-crossing Dyck path give us an enumeration of realizers. An algorithm generating p non-crossing Dyck paths, is also proposed. It allows us to generate randomly realizers in linear time. Then, we show that thanks to realizers, we can draw plane graphs with polylines on grids of optimal width and area. Finally, we propose a generalization of minimal realizers to connected planar graphs : well-orderly spanning trees. Using this generalization and with a particular triangulation algorithm, we present a new 5.007n bit planar graph encoding

Baxter permutations and plane bipolar orientations

We present a simple bijection between Baxter permutations of size $n$ and plane bipolar orientations with n edges. This bijection translates several classical parameters of permutations (number of ascents, right-to-left maxima, left-to-right minima...) into natural parameters of plane bipolar orientations (number of vertices, degree of the sink, degree of the source...), and has remarkable symmetry properties. By specializing it to Baxter permutations avoiding the pattern 2413, we obtain a bijection with non-separable planar maps. A further specialization yields a bijection between permutations avoiding 2413 and 3142 and series-parallel maps.Comment: 22 page

Upper and Lower Bounds for Competitive Online Routing on Delaunay Triangulations

Consider a weighted graph G where vertices are points in the plane and edges are line segments. The weight of each edge is the Euclidean distance between its two endpoints. A routing algorithm on G has a competitive ratio of c if the length of the path produced by the algorithm from any vertex s to any vertex t is at most c times the length of the shortest path from s to t in G. If the length of the path is at most c times the Euclidean distance from s to t, we say that the routing algorithm on G has a routing ratio of c.We present an online routing algorithm on the Delaunay triangulation with competitive and routing ratios of 5.90. This improves upon the best known algorithm that has competitive and routing ratio 15.48. The algorithm is a generalization of the deterministic 1-local routing algorithm by Chew on the L1-Delaunay triangulation. When a message follows the routing path produced by our algorithm, its header need only contain the coordinates of s and t. This is an improvement over the currently known competitive routing algorithms on the Delaunay triangulation, for which the header of a message must additionally contain partial sums of distances along the routing path.We also show that the routing ratio of any deterministic k-local algorithm is at least 1.70 for the Delaunay triangulation and 2.70 for the L1-Delaunay triangulation. In the case of the L1-Delaunay triangulation, this implies that even though there exists a path between two points x and y whose length is at most 2.61|[xy]| (where |[xy]| denotes the length of the line segment [xy]), it is not always possible to route a message along a path of length less than 2.70|[xy]|. From these bounds on the routing ratio, we derive lower bounds on the competitive ratio of 1.23 for Delaunay triangulations and 1.12 for L1-Delaunay triangulations

Catalan's intervals and realizers of triangulations

The Stanley lattice, Tamari lattice and Kreweras lattice are three remarkable orders defined on the set of Catalan objects of a given size. These lattices are ordered by inclusion: the Stanley lattice is an extension of the Tamari lattice which is an extension of the Kreweras lattice. The Stanley order can be defined on the set of Dyck paths of size n as the relation of being above. Hence, intervals in the Stanley lattice are pairs of non-crossing Dyck paths. In a former article, the second author defined a bijection Φ between pairs of non-crossing Dyck paths and the realizers of triangulations (or Schnyder woods). We give a simpler description of the bijection Φ. Then, we study the restriction of Φ to Tamari's and Kreweras' intervals. We prove that Φ induces a bijection between Tamari intervals and minimal realizers. This gives a bijection between Tamari intervals and triangulations. We also prove that Φ induces a bijection between Kreweras intervals and the (unique) realizers of stack triangulations. Thus, Φ induces a bijection between Kreweras intervals and stacktriangulations which are known to be in bijection with ternary trees

Modeling and Practical Evaluation of a Service Location Problem in Large Scale Networks

International audienceWe consider a generalization of a classical optimization problem related to server and replica location problems in networks. More precisely, we suppose that a set of users distributed over a network wish to have access to a particular service proposed by a set of providers. The aim is then to distinguish a set of service providers able to offer a sufficient amount of resources in order to satisfy the requests of the clients. Moreover, a quality of service following some requirements in terms of latencies is desirable. A smart repartition of the servers in the network may also ensure good fault tolerance properties. We model this problem as a variant of Bin Packing, namely Bin Packing under Distance Constraint (BPDC) where the goal is to build a minimal number of bins (i.e. to choose a minimal number of servers) so that (i) each client is associated to exactly one server, (ii) the capacity of the server is large enough to satisfy the requests of its clients and (iii) the distance between two clients associated to the same server is minimized. We prove that this problem is hard to approximate even when using resource augmentation techniques : we compare the number of obtained bins when using polynomial time algorithms allowed to build bins of diameter at most b*dmax, for b>1, to the optimal number of bins of diameter at most dmax. On the one hand, we prove that (i) if b=(2-e), BPDC is hard to approximate within any constant approximation ratio, for any e>0; and that (ii) BPDC is hard to approximate at a ratio lower than 3/2 even if resource augmentation is used. On the other hand, if b=2, we propose a polynomial time approximation algorithm for BPDC with approximation ratio 7/3 in the general case. We show how to turn an approximation algorithm for BPDC into an approximation algorithm for the non-uniform capacitated K-center problem and vice-versa. Then, we present a comparison of the quality of results for BPDC in the context of several Internet latency embedding tools such as Sequoia and Vivaldi, using datasets based on PlanetLab latency measurements.Nous considérons une généralisation d'un problème d'optimisation classique lié au placement de serveurs et de réplicats dans les réseaux. Plus précisément, nous supposons qu'un ensemble d'utilisateurs au sein d'un réseau souhaite accéder à un service particulier proposé par un ensemble de fournisseurs de ce service. L'objectif est alors d'identifier un ensemble de fournisseurs de service capable d'offrir suffisamment de ressources pour répondre aux requêtes des clients. Par ailleurs, une certaine qualité de service relativement aux temps de communications est désirable. Une répartition judicieuse des serveurs dans le réseau offrirait également de bonnes propriétés de tolérance aux pannes. Nous modélisons ce problème comme une variante de Bin Packing, le Bin Packing avec Contrainte de Distance (BPDC en anglais) où le but est de construire un minimum de groupes (i.e. de choisir un nombre minimal de serveurs) de telle sorte que (i) chaque client est associé à exactement un serveur, (ii) la capacité dudit serveur est suffisante pour répondre aux requêtes des clients qui lui sont associés et (iii) la distance entre deux clients associés au même serveur est minimisée. Nous prouvons que ce problème est inapproximable même en utilisant des techniques d'augmentation de ressources : le nombre de groupes obtenus en utilisant des algorithmes s'exécutant en temps polynomial et autorisés à construire des groupes de diamètre au plus b*dmax, avec b>1, est comparé au nombre de groupes d'une solution optimale construisant des groupes de diamètre au plus dmax. D'un côté, nous prouvons que (i) si b=(2-e), BPDC est inapproximable à facteur constant, pour tout e>0; et que (ii) BPDC est inapproximableà un facteur inférieur à 3/2 même en utilisant de l'augmentation de ressources. D'un autre côté, si b=2, nous proposons un algorithme s'exécutant en temps polynomial pour BPDC assurant un facteur d'approximation de 7/3 dans le cas général. Nous montrons également comment transformer un algorithme d'approximation pour BPDC en un algorithme d'approximation pour le K-centre non uniforme avec capacités, et vice-versa. Enfin, nous présentons une comparaison qualitative de nos résultats pour BPDC en utilisant plusieurs outils de plongement de l'espace des latences d'Internet, comme Sequoia et Vivaldi