USING COMBINATORIAL MAPS FOR ALGORITHMS ON GRAPHS

The aim of this paper is to come back to a data structure representation of graph by permutations. This originated in the years 1960-1970 by contributions due to J. Edmonds [7], A. Jacques [11], W. Tutte [22] in order to consider the embedding of a graph in a surface as a combinatorial object. Some algebraic developments where suggested in [4] and [12]. It was also used for implementation in different situation, like planarity testing by H. de Fraysseix and P. Rosenstiehl [6], computer vision by G. Damiand and A. Dupas [5] or formal proofs by G. Gonthier [9].

Connected Chord Diagrams and Bridgeless Maps

We present a surprisingly new connection between two well-studied combinatorial classes: rooted connected chord diagrams on one hand, and rooted bridgeless combinatorial maps on the other hand. We describe a bijection between these two classes, which naturally extends to indecomposable diagrams and general rooted maps. As an application, this bijection provides a simplifying framework for some technical quantum field theory work realized by some of the authors. Most notably, an important but technical parameter naturally translates to vertices at the level of maps. We also give a combinatorial proof to a formula which previously resulted from a technical recurrence, and with similar ideas we prove a conjecture of Hihn. Independently, we revisit an equation due to Arquès and Béraud for the generating function counting rooted maps with respect to edges and vertices, giving a new bijective interpretation of this equation directly on indecomposable chord diagrams, which moreover can be specialized to connected diagrams and refined to incorporate the number of crossings. Finally, we explain how these results have a simple application to the combinatorics of lambda calculus, verifying the conjecture that a certain natural family of lambda terms is equinumerous with bridgeless maps.

The joint distribution of the marginals of multipartite random quantum states

We study the joint distribution of the set of all marginals of a random Wishart matrix acting on a tensor product Hilbert space. We compute the limiting free mixed cumulants of the marginals, and we show that in the balanced asymptotical regime, the marginals are asymptotically free. We connect the matrix integrals relevant to the study of operators on tensor product spaces with the corresponding classes of combinatorial maps, for which we develop the combinatorial machinery necessary for the asymptotic study. Finally, we present some applications to the theory of random quantum states in quantum information theory.

Distributed Combinatorial Maps for Parallel Mesh Processing

We propose a new strategy for the parallelization of mesh processing algorithms. Our main contribution is the definition of distributed combinatorial maps (called n-dmaps), which allow us to represent the topology of big meshes by splitting them into independent parts. Our mathematical definition ensures the global consistency of the meshes at their interfaces. Thus, an n-dmap can be used to represent a mesh, to traverse it, or to modify it by using different mesh processing algorithms. Moreover, an nD mesh with a huge number of elements can be considered, which is not possible with a sequential approach and a regular data structure. We illustrate the interest of our solution by presenting a parallel adaptive subdivision method of a 3D hexahedral mesh, implemented in a distributed version. We report space and time performance results that show the interest of our approach for parallel processing of huge meshes.

EXTENDED MAPTREE: A REPRESENTATION OF FINE-GRAINED TOPOLOGY AND SPATIAL HIERARCHY OF BIM

Spatial queries play significant roles in exchanging Building Information Modeling (BIM) data and integrating BIM with indoor spatial information. However, topological operators implemented for BIM spatial queries are limited to qualitative relations (e.g. touching, intersecting). To overcome this limitation, we propose an extended maptree model to represent the fine-grained topology and spatial hierarchy of indoor spaces. The model is based on a maptree which consists of combinatorial maps and an adjacency tree. Topological relations (e.g., adjacency, incidence, and covering) derived from BIM are represented explicitly and formally by extended maptrees, which can facilitate the spatial queries of BIM. To construct an extended maptree, we first use a solid model represented by vertical extrusion and boundary representation to generate the isolated 3-cells of combinatorial maps. Then, the spatial relationships defined in IFC are used to sew them together. Furthermore, the incremental edges of extended maptrees are labeled as removed 2-cells. Based on this, we can merge adjacent 3-cells according to the spatial hierarchy of IFC.

Colored Triangulations of Arbitrary Dimensions are Stuffed Walsh Maps

Regular edge-colored graphs encode colored triangulations of pseudo-manifolds. Here we study families of edge-colored graphs built from a finite but arbitrary set of building blocks, which extend the notion of $p$-angulations to arbitrary dimensions. We prove the existence of a bijection between any such family and some colored combinatorial maps which we call stuffed Walsh maps. Those maps generalize Walsh's representation of hypermaps as bipartite maps, by replacing the vertices which correspond to hyperedges with non-properly-edge-colored maps. This shows the equivalence of tensor models with multi-trace, multi-matrix models by extending the intermediate field method perturbatively to any model. We further use the bijection to study the graphs which maximize the number of faces at fixed number of vertices and provide examples where the corresponding stuffed Walsh maps can be completely characterized.

EXTRACTION OF THE 3D FREE SPACE FROM BUILDING MODELS FOR INDOOR NAVIGATION

For several decades, indoor navigation has been exclusively investigated in a 2D perspective, based on floor plans, projection and other 2D representations of buildings. Nevertheless, 3D representations are closer to our reality and offer a more intuitive description of the space configuration. Thanks to recent advances in 3D modelling, 3D navigation is timidly but increasingly gaining in interest through the indoor applications. But, because the structure of indoor environment is often more complex than outdoor, very simplified models are used and obstacles are not considered for indoor navigation leading to limited possibilities in complex buildings. In this paper we consider the entire configuration of the indoor environment in 3D and introduce a method to extract from it the actual navigable space as a network of connected 3D spaces (volumes). We describe how to construct such 3D free spaces from semantically rich and furnished IFC models. The approach combines the geometric, the topological and the semantic information available in a 3D model to isolate the free space from the rest of the components. Furthermore, the extraction of such navigable spaces in building models lacking of semantic information is also considered. A data structure named combinatorial maps is used to support the operations required by the process while preserving the topological and semantic information of the input models.

STORING A 3D CITY MODEL, ITS LEVELS OF DETAIL AND THE CORRESPONDENCES BETWEEN OBJECTS AS A 4D COMBINATORIAL MAP

3D city models of the same region at multiple LODs are encumbered by the lack of links between corresponding objects across LODs. In practice, this causes inconsistency during updates and maintenance problems. A radical solution to this problem is to model the LOD of a model as a dimension in the geometric sense, such that a set of connected polyhedra at a series of LODs is modelled as a single polychoron—the 4D analogue of a polyhedron. This approach is generally used only conceptually and then discarded at the implementation stage, losing many of its potential advantages in the process. This paper therefore shows that this approach can be instead directly realised using 4D combinatorial maps, making it possible to store all topological relationships between objects.