Generic method for bijections between blossoming trees and planar maps

This article presents a unified bijective scheme between planar maps and blossoming trees, where a blossoming tree is defined as a spanning tree of the map decorated with some dangling half-edges that enable to reconstruct its faces. Our method generalizes a previous construction of Bernardi by loosening its conditions of applications so as to include annular maps, that is maps embedded in the plane with a root face different from the outer face. The bijective construction presented here relies deeply on the theory of \alpha-orientations introduced by Felsner, and in particular on the existence of minimal and accessible orientations. Since most of the families of maps can be characterized by such orientations, our generic bijective method is proved to capture as special cases all previously known bijections involving blossoming trees: for example Eulerian maps, m-Eulerian maps, non separable maps and simple triangulations and quadrangulations of a k-gon. Moreover, it also permits to obtain new bijective constructions for bipolar orientations and d-angulations of girth d of a k-gon. As for applications, each specialization of the construction translates into enumerative by-products, either via a closed formula or via a recursive computational scheme. Besides, for every family of maps described in the paper, the construction can be implemented in linear time. It yields thus an effective way to encode and generate planar maps. In a recent work, Bernardi and Fusy introduced another unified bijective scheme, we adopt here a different strategy which allows us to capture different bijections. These two approaches should be seen as two complementary ways of unifying bijections between planar maps and decorated trees.Comment: 45 pages, comments welcom

Agglomerative Clustering of Growing Squares

We study an agglomerative clustering problem motivated by interactive glyphs in geo-visualization. Consider a set of disjoint square glyphs on an interactive map. When the user zooms out, the glyphs grow in size relative to the map, possibly with different speeds. When two glyphs intersect, we wish to replace them by a new glyph that captures the information of the intersecting glyphs. We present a fully dynamic kinetic data structure that maintains a set of $n$ disjoint growing squares. Our data structure uses $O(n (\log n \log\log n)^2)$ space, supports queries in worst case $O(\log^3 n)$ time, and updates in $O(\log^7 n)$ amortized time. This leads to an $O(n\alpha(n)\log^7 n)$ time algorithm to solve the agglomerative clustering problem. This is a significant improvement over the current best $O(n^2)$ time algorithms.Comment: 14 pages, 7 figure

Transversal structures on triangulations: a combinatorial study and straight-line drawings

This article focuses on a combinatorial structure specific to triangulated plane graphs with quadrangular outer face and no separating triangle, which are called irreducible triangulations. The structure has been introduced by Xin He under the name of regular edge-labelling and consists of two bipolar orientations that are transversal. For this reason, the terminology used here is that of transversal structures. The main results obtained in the article are a bijection between irreducible triangulations and ternary trees, and a straight-line drawing algorithm for irreducible triangulations. For a random irreducible triangulation with $n$ vertices, the grid size of the drawing is asymptotically with high probability $11n/27\times 11n/27$ up to an additive error of \cO(\sqrt{n}). In contrast, the best previously known algorithm for these triangulations only guarantees a grid size $(\lceil n/2\rceil -1)\times \lfloor n/2\rfloor$.Comment: 42 pages, the second version is shorter, focusing on the bijection (with application to counting) and on the graph drawing algorithm. The title has been slightly change

Identification of a homolog of Arabidopsis DSP4 (SEX4) in chestnut: its induction and accumulation in stem amyloplasts during winter or in response to the cold_

Oligosaccharide synthesis is an important cryoprotection strategy used by woody plants during winter dormancy. At the onset of autumn, starch stored in the stem and buds is broken down in response to the shorter days and lower temperatures resulting in the buildup of oligosaccharides. Given that the enzyme DSP4 is necessary for diurnal starch degradation in Arabidopsis leaves, this study was designed to address the role of DSP4 in this seasonal process in Castanea sativa Mill. The expression pattern of the CsDSP4 gene in cells of the chestnut stem was found to parallel starch catabolism. In this organ, DSP4 protein levels started to rise at the start of autumn and elevated levels persisted until the onset of spring. In addition, exposure of chestnut plantlets to 4 °C induced the expression of the CsDSP4 gene. In dormant trees or cold-stressed plantlets, the CsDSP4 protein was immunolocalized both in the amyloplast stroma and nucleus of stem cells, whereas in the conditions of vegetative growth, immunofluorescence was only detected in the nucleus. The studies indicate a potential role for DSP4 in starch degradation and cold acclimation following low temperature exposure during activity–dormancy transition

Fast Generation of Random Spanning Trees and the Effective Resistance Metric

We present a new algorithm for generating a uniformly random spanning tree in an undirected graph. Our algorithm samples such a tree in expected $\tilde{O}(m^{4/3})$ time. This improves over the best previously known bound of $\min(\tilde{O}(m\sqrt{n}),O(n^{\omega}))$ -- that follows from the work of Kelner and M\k{a}dry [FOCS'09] and of Colbourn et al. [J. Algorithms'96] -- whenever the input graph is sufficiently sparse. At a high level, our result stems from carefully exploiting the interplay of random spanning trees, random walks, and the notion of effective resistance, as well as from devising a way to algorithmically relate these concepts to the combinatorial structure of the graph. This involves, in particular, establishing a new connection between the effective resistance metric and the cut structure of the underlying graph