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163
Random planar lattices and integrated superBrownian excursion
 Probab. Th. Rel. Fields
"... Abstract. In this paper, a surprising connection is described between a specific brand of random lattices, namely planar quadrangulations, and Aldous’ Integrated SuperBrownian Excursion (ISE). As a consequence, the radius rn of a random quadrangulation with n faces is shown to converge, up to scalin ..."
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Cited by 88 (3 self)
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Abstract. In this paper, a surprising connection is described between a specific brand of random lattices, namely planar quadrangulations, and Aldous’ Integrated SuperBrownian Excursion (ISE). As a consequence, the radius rn of a random quadrangulation with n faces is shown to converge, up to scaling, to the width r = R−L of the support of the onedimensional ISE, or precisely: n −1/4 rn law − → (8/9) 1/4 r. More generally the distribution of distances to a random vertex in a random quadrangulation is described in its scaled limit by the random measure ISE shifted to set the minimum of its support in zero. The first combinatorial ingredient is an encoding of quadrangulations by trees embedded in the positive halfline, reminiscent of Cori and Vauquelin’s well labelled trees. The second step relates these trees to embedded (discrete) trees in the sense of Aldous, via the conjugation of tree principle, an analogue for trees of Vervaat’s construction of the Brownian excursion from the bridge. From probability theory, we need a new result of independent interest: the weak convergence of the encoding of a random embedded plane tree by two contour walks (e (n) , ˆ W (n) ) to the Brownian snake description (e, ˆ W) of ISE. Our results suggest the existence of a Continuum Random Map describing in term of ISE the scaled limit of the dynamical triangulations considered in twodimensional pure quantum gravity. 1.
Recent advances in compression of 3D meshes
 In Advances in Multiresolution for Geometric Modelling
, 2003
"... Summary. 3D meshes are widely used in graphic and simulation applications for approximating 3D objects. When representing complex shapes in a raw data format, meshes consume a large amount of space. Applications calling for compact storage and fast transmission of 3D meshes have motivated the multit ..."
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Cited by 81 (3 self)
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Summary. 3D meshes are widely used in graphic and simulation applications for approximating 3D objects. When representing complex shapes in a raw data format, meshes consume a large amount of space. Applications calling for compact storage and fast transmission of 3D meshes have motivated the multitude of algorithms developed to efficiently compress these datasets. In this paper we survey recent developments in compression of 3D surface meshes. We survey the main ideas and intuition behind techniques for singlerate and progressive mesh coding. Where possible, we discuss the theoretical results obtained for asymptotic behavior or optimality of the approach. We also list some open questions and directions for future research. 1
The topological structure of scaling limits of large planar maps
 Invent. Math
"... We discuss scaling limits of large bipartite planar maps. If p ≥ 2 is a fixed integer, we consider, for every integer n ≥ 2, a random planar map Mn which is uniformly distributed over the set of all rooted 2pangulations with n faces. Then, at least along a suitable subsequence, the metric space con ..."
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Cited by 75 (12 self)
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We discuss scaling limits of large bipartite planar maps. If p ≥ 2 is a fixed integer, we consider, for every integer n ≥ 2, a random planar map Mn which is uniformly distributed over the set of all rooted 2pangulations with n faces. Then, at least along a suitable subsequence, the metric space consisting of the set of vertices of Mn, equipped with the graph distance rescaled by the factor n −1/4, converges in distribution as n → ∞ towards a limiting random compact metric space, in the sense of the GromovHausdorff distance. We prove that the topology of the limiting space is uniquely determined independently of p and of the subsequence, and that this space can be obtained as the quotient of the Continuum Random Tree for an equivalence relation which is defined from Brownian labels attached to the vertices. We also verify that the Hausdorff dimension of the limit is almost surely equal to 4. 1
Spatial networks
 PHYSICS REPORTS
, 2010
"... Complex systems are very often organized under the form of networks where nodes and edges are embedded in space. Transportation and mobility networks, Internet, mobile phone networks, power grids, social and contact networks, neural networks, are all examples where space is relevant and where topolo ..."
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Cited by 70 (5 self)
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Complex systems are very often organized under the form of networks where nodes and edges are embedded in space. Transportation and mobility networks, Internet, mobile phone networks, power grids, social and contact networks, neural networks, are all examples where space is relevant and where topology alone does not contain all the information. Characterizing and understanding
Bijective Census and Random Generation of Eulerian Planar Maps with Prescribed Vertex Degrees
, 1997
"... We give a bijection between Eulerian planar maps with prescribed vertex degrees, and some plane trees that we call balanced Eulerian trees. To enumerate the latter, we introduce conjugation classes of planted plane trees. In particular the result answers a question of Bender and Canfield in [BC94] a ..."
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Cited by 52 (5 self)
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We give a bijection between Eulerian planar maps with prescribed vertex degrees, and some plane trees that we call balanced Eulerian trees. To enumerate the latter, we introduce conjugation classes of planted plane trees. In particular the result answers a question of Bender and Canfield in [BC94] and allows uniform random generation of Eulerian planar maps with restricted vertex degrees. Using a well known correspondence between 4regular planar maps with n vertices and planar maps with n edges we obtain an algorithm to generate uniformly such maps with complexity O(n). Our bijection is also refined to give a combinatorial interpretation of a parameterization of Arques ([Arq87]) of the generating function of planar maps with respect to vertices and faces.
Random maps, coalescing saddles, singularity analysis, and Airy phenomena
 Random Structures & Algorithms
, 2001
"... A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponentialquadratic type, that is, Gaussian. We exhibit a class of "universal" phenomena that are of the exponentialcubic type, corresponding to distributions that i ..."
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Cited by 52 (6 self)
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A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponentialquadratic type, that is, Gaussian. We exhibit a class of "universal" phenomena that are of the exponentialcubic type, corresponding to distributions that involve the Airy function. In this paper, such Airy phenomena are related to the coalescence of saddle points and the confluence of singularities of generating functions. For about a dozen types of random planar maps, a common Airy distribution (equivalently, a stable law of exponent 3/2) describes the sizes of cores and of largest (multi)connected components. Consequences include the analysis and fine optimization of random generation algorithms for multiply connected planar graphs. Based on an extension of the singularity analysis framework suggested by the Airy case, the paper also presents a general classification of compositional schemas in analytic combinatorics.
Short Encodings of Planar Graphs and Maps
 Discrete Applied Mathematics
, 1993
"... We discuss spaceefficient encoding schemes for planar graphs and maps. Our results improve on the constants of previous schemes and can be achieved with simple encoding algorithms. They are nearoptimal in number of bits per edge. 1 Introduction In this paper we discuss spaceefficient binary enco ..."
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Cited by 41 (0 self)
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We discuss spaceefficient encoding schemes for planar graphs and maps. Our results improve on the constants of previous schemes and can be achieved with simple encoding algorithms. They are nearoptimal in number of bits per edge. 1 Introduction In this paper we discuss spaceefficient binary encoding schemes for several classes of unlabeled connected planar graphs and maps. In encoding a graph we must encode the incidences among vertexes and edges. By maps we understand topological equivalence classes of planar embeddings of planar graphs. In encoding a map we are required to encode the topology of the embedding i.e., incidences among faces, edges, and vertexes, as well as the graph. Each map is an embedding of a unique graph, but a given graph may have multiple embeddings. Hence maps must require more bits to encode than graphs in some average sense. There are a number of recent results on spaceefficient encoding. A standard adjacency list encoding of an unlabeled graph G requires...
The number of labeled 2connected planar graphs
 Journal of Combinatorics
, 2000
"... We derive the asymptotic expression for the number of labeled 2connected planar graphs with respect to vertices and edges. We also show that almost all such graphs with n vertices contain many copies of any fixed planar graph, and this implies that almost all such graphs have large automorphism gro ..."
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Cited by 41 (2 self)
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We derive the asymptotic expression for the number of labeled 2connected planar graphs with respect to vertices and edges. We also show that almost all such graphs with n vertices contain many copies of any fixed planar graph, and this implies that almost all such graphs have large automorphism groups.
The Random Planar Graph
 Congressus Numerantium
, 1996
"... We construct a Markov chain whose stationary distribution is uniform over all planar subgraphs of a graph. In the case of the complete graph our experiments suggest that the random simple planar graph on n vertices is connected but not 2connected and has approximately 2n edges. We present a rs ..."
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Cited by 38 (1 self)
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We construct a Markov chain whose stationary distribution is uniform over all planar subgraphs of a graph. In the case of the complete graph our experiments suggest that the random simple planar graph on n vertices is connected but not 2connected and has approximately 2n edges. We present a rst attack on the problem of describing what the random planar graph looks like.