A bijection for rooted maps on general surfaces

We extend the Marcus-Schaeffer bijection between orientable rooted bipartite quadrangulations (equivalently: rooted maps) and orientable labeled one-face maps to the case of all surfaces, that is orientable and non-orientable as well. This general construction requires new ideas and is more delicate than the special orientable case, but it carries the same information. In particular, it leads to a uniform combinatorial interpretation of the counting exponent $\frac{5(h-1)}{2}$ for both orientable and non-orientable rooted connected maps of Euler characteristic $2-2h$, and of the algebraicity of their generating functions, similar to the one previously obtained in the orientable case via the Marcus-Schaeffer bijection. It also shows that the renormalization factor $n^{1/4}$ for distances between vertices is universal for maps on all surfaces: the renormalized profile and radius in a uniform random pointed bipartite quadrangulation on any fixed surface converge in distribution when the size $n$ tends to infinity. Finally, we extend the Miermont and Ambj{\o}rn-Budd bijections to the general setting of all surfaces. Our construction opens the way to the study of Brownian surfaces for any compact 2-dimensional manifold.Comment: v2: 55 pages, 22 figure

On the Influence of the instance structure for metaheuristic performances -- Application to a graph drawing problem

International audienceMetaheuristics are now so common that for some classical hard combinatorial problems, there exist more than ten variants. Thus, the issue of comparing optimization methods is crucial. In this paper, we focus on one aspect of this question: the impact of the choice of the test instances on the metaheuristic performances and the possible link with the fitness landscape structure. We base our experimental framework on the arc crossing minimization problem for layered digraphs. We compare a hybridized genetic algorithm and a multistart descent which are among the best approaches to this problem. We worked on two instance families with various sizes and structural complexities: small graphs which are easy to draw on a standard size support, and large graphs specifically built for our experiments. We show that, for the smallest instances, there is no significant difference between methods whereas for graphs similar to those classically used nowadays in applications the genetic algorithm is better, and for the largest graphs (with a scaling factor up to $10^{300}$), the multistart descent is the best method. These results suggest that for ``structured'' fitness landscapes associated with real-life instances the GA exploits its implicit learning. On the other hand for very large landscapes with probably numerous local optima, only one exploration on a larger scale can be provided by local searches from a random starting point, cheap in computing effort

Compact Hierarchical Graph Drawings via Quadratic Layer Assignment

We propose a new mixed-integer programming formulation that very naturally expresses the layout restrictions of a layered (hierarchical) graph drawing and several associated objectives, such as a minimum total arc length, number of reversed arcs, and width, or the adaptation to a specific drawing area, as a special quadratic assignment problem. Our experiments show that it is competitive to another formulation that we slightly simplify as well

Layout of compound directed graphs

We present a method for the layout of compound directed graphs that is based on the hierarchical layer layout method. Our method has similarities with the method of Sugiyama and Misue (IEEE Trans. Sys., Man, Cybernetics, 21(4), pp. 876-892, 1991) but gives different results: It uses a global partitioning into layers and tries to produce placements of nodes such that border rectangles can be drawn around the nodes of each subgraph. The method is implemented in the VCG tool

A bijection for rooted maps on general surfaces (extended abstract)

International audienceWe extend the Marcus-Schaeffer bijection between orientable rooted bipartite quadrangulations (equivalently: rooted maps) and orientable labeled one-face maps to the case of all surfaces, orientable or non-orientable. This general construction requires new ideas and is more delicate than the special orientable case, but carries the same information. It thus gives a uniform combinatorial interpretation of the counting exponent $\frac{5(h-1)}{2}$ for both orientable and non-orientable maps of Euler characteristic $2-2h$ and of the algebraicity of their generating functions. It also shows the universality of the renormalization factor $n$¼ for the metric of maps, on all surfaces: the renormalized profile and radius in a uniform random pointed bipartite quadrangulation of size $n$ on any fixed surface converge in distribution. Finally, it also opens the way to the study of Brownian surfaces for any compact 2-dimensional manifold.Nous étendons la bijection de Marcus et Schaeffer entre quadrangulations biparties orientables (de manière équivalente: cartes enracinées) et cartes à une face étiquetées orientables à toutes les surfaces, orientables ou non. Cette construction générale requiert des idées nouvelles et est plus délicate que dans le cas particulier orientable, mais permet des utilisations similaires. Elle donne donc une interprétation combinatoire uniforme de l’exposant de comptage $\frac{5(h-1)}{2}$ pour les cartes orientables et non-orientables de caractéristique d’Euler $2-2h$, et de l’algébricité des fonctions génératrices. Elle montre l’universalité du facteur de normalisation $n$¼ pour la métrique des cartes, sur toutes les surfaces: le profil et le rayon d’une quadrangulation enracinée pointée sur une surface fixée converge en distribution. Enfin, elle ouvre à la voie à l’étude des surfaces Browniennes pour toute 2-variété compacte

A generic algorithm for layout of biological networks

BackgroundBiological networks are widely used to represent processes in biological systems and to capture interactions and dependencies between biological entities. Their size and complexity is steadily increasing due to the ongoing growth of knowledge in the life sciences. To aid understanding of biological networks several algorithms for laying out and graphically representing networks and network analysis results have been developed. However, current algorithms are specialized to particular layout styles and therefore different algorithms are required for each kind of network and/or style of layout. This increases implementation effort and means that new algorithms must be developed for new layout styles. Furthermore, additional effort is necessary to compose different layout conventions in the same diagram. Also the user cannot usually customize the placement of nodes to tailor the layout to their particular need or task and there is little support for interactive network exploration.ResultsWe present a novel algorithm to visualize different biological networks and network analysis results in meaningful ways depending on network types and analysis outcome. Our method is based on constrained graph layout and we demonstrate how it can handle the drawing conventions used in biological networks.ConclusionThe presented algorithm offers the ability to produce many of the fundamental popular drawing styles while allowing the exibility of constraints to further tailor these layouts.publishe

Minimum-width graph layering revisited

The minimum-width layering problem tackles one step of a layout pipeline to create top-down drawings of directed acyclic graphs. Thereby, it aims at keeping the overall width of the drawing small. We study layering heuristics for this problem as presented by Nikolov et al. as well as traditional layering methods with respect to the questions how close they are to the true minimal width and how well they perform in practice, especially, if a particular drawing area is prescribed. We find that when applied carefully and at the right moment the layering heuristics can, compared to traditional layering methods, produce better layerings for prescribed drawing areas and for graphs with varying node dimensions. Still, we also find that there is room for improvement

Wrapping layered graphs

We present additions to the widely-used layout method for directed acyclic graphs of Sugiyama et al. [16] that allow to better utilize a prescribed drawing area. The method itself partitions the graph's nodes into layers. When drawing from top to bottom, the number of layers directly impacts the height of a resulting drawing and is bound from below by the graph's longest path. As a consequence, the drawings of certain graphs are significantly taller than wide, making it hard to properly display them on a medium such as a computer screen without scaling the graph's elements down to illegibility. We address this with the Wrapping Layered Graphs Problem (WLGP), which seeks for cut indices that split a given layering into chunks that are drawn side-by-side with a preferably small number of edges wrapping backwards. Our experience and a quantitative evaluation indicate that the proposed wrapping allows an improved presentation of narrow graphs, which occur frequently in practice and of which the internal compiler representation SCG is one example