Gilles Schaeffer

Optimal coding and sampling of triangulations

We present a bijection between the set of plane triangulations (aka maximal planar graphs) and a simple subset of the set of plane trees with two leaves adjacent to each node. The construction takes advantage of Schnyder tree decompositions of plane triangulations. This bijection yields an interpretation of the formula for the number of plane triangulations with n vertices. Moreover, the construction is simple enough to induce a linear random sampling algorithm, and an explicit information theory optimal encoding. Finally, we extend our bijection approach to triangulations of a polygon with k sides with m inner vertices, and develop in passing new results about Schnyder tree decompositions for these objects.

Boltzmann Samplers for the Random Generation of Combinatorial Structures

This article proposes a surprisingly simple framework for the random generation of combinatorial configurations based on what we call Boltzmann models. The idea is to perform random generation of possibly complex structured objects by placing an appropriate measure spread over the whole of a combinatorial class – an object receives a probability essentially proportional to an exponential of its size. As demonstrated here, the resulting algorithms based on real-arithmetic operations often operate in linear time. They can be implemented easily, be analysed mathematically with great precision, and, when suitably tuned, tend to be very efficient in practice.

A combinatorial approach to jumping particles

In this paper we consider a model of particles jumping on a row of cells, called in physics the one-dimensional totally asymmetric exclusion process (TASEP). More precisely, we deal with the TASEP with open or periodic boundary conditions and with two or three types of particles. From the point of view of combinatorics a remarkable feature of this Markov chain is that it involves Catalan numbers in several entries of its stationary distribution.We give a combinatorial interpretation and a simple proof of these observations. In doing this we reveal a second row of cells, which is used by particles to travel backward. As a byproduct we also obtain an interpretation of the occurrence of the Brownian excursion in the description of the density of particles on a long row of cells.

Random sampling of large planar maps and convex polyhedra

We present a versatile almost uniform random generator for planar embeddings of graphs (usually called \emph{maps}). This simple pseudo-algorithm is linear on average and gives in a few seconds random maps with up to one million edges or vertices. The class of planar graphs that can be obtained includes graphs of convex polyhedra (or 3-connected planar graphs) and convex irreducible triangulations (or 4-connected maximal planar graphs). Our algorithm relies on a combinatorial approach. First, new simpler compact encodings are defined using canonical unlabelled covering trees. Second, a general \emph{extraction/rejection} pseudo-algorithm is defined for composed structures. If correctly \emph{tuned} (we provide the necessary analysis for maps), it applies efficiently to a much wider class of planar graphs than previously known methods.

Succinct representation of triangulations with a boundary

We consider the problem of designing succinct geometric data structures while maintaining efficient navigation operations. A data structure is said succinct if the asymptotic amount of space it uses matches the entropy of the class of structures represented. For the case of planar triangulations with a boundary we propose a succinct representation of the combinatorial information that improves to 2.175 bits per triangle the asymptotic amount of space required and that supports the navigation between adjacent triangles in constant time (as well as other standard operations). For triangulations with m faces of a surface with genus g, our representation requires asymptotically an extra amount of 36(g - 1)lg m bits (which is negligible as long as g≪m/lg m).

A Bijective Census of Nonseparable Planar Maps

Bijections are obtained between nonseparable planar maps and two different kinds of trees: description trees and skew ternary trees. A combinatorial relation between the latter and ternary trees allows bijective enumeration and random generation of nonseparable planar maps. The involved bijections take account of the usual combinatorial parameters and give a bijective proof of formulae established by Brown and Tutte. These results, combined with a bijection due to Goulden and West, give a purely combinatorial enumeration of two-stack-sortable permutations.

A bijective proof of Jackson's formula for the number of factorizations of a cycle

Factorizations of the cyclic permutation (12...N) into two permutations with respectively n and m cycles, or, equivalently, unicellular bicolored maps with N edges and n white and m black vertices, have been enumerated independantly by Jackson and Adrianov using evaluations of characters of the symmetric group. In this paper we present a bijection between unicellular partitioned bicolored maps and couples made of an ordered bicolored tree and a partial permutation, that allows for a combinatorial derivation of these results. Our work is closely related to a recent construction of Goulden and Nica for the celebrated Harer-Zagier formula, and indeed we provide a unified presentation of both bijections in terms of Eulerian tours in graphs.

Optimal succinct representations of planar maps

This paper addresses the problem of representing the connectivity information of geometric objects using as little memory as possible. As opposed to raw compression issues, the focus is here on designing data structures that preserve the possibility of answering incidence queries in constant time. We propose in particular the first optimal representations for 3-connected planar graphs and triangulations, which are the most standard classes of graphs underlying meshes with spherical topology. Optimal means that these representations asymptotically match the respective entropy of the two classes, namely 2 bits per edge for 3-connected planar graphs, and 1.62 bits per triangle or equivalently 3.24 bits per vertex for triangulations.

Succinct representations of planar maps

This paper addresses the problem of representing the connectivity information of geometric objects, using as little memory as possible. As opposed to raw compression issues, the focus here is on designing data structures that preserve the possibility of answering incidence queries in constant time. We propose, in particular, the first optimal representations for 3-connected planar graphs and triangulations, which are the most standard classes of graphs underlying meshes with spherical topology. Optimal means that these representations asymptotically match the respective entropy of the two classes, namely 2 bits per edge for 3-connected planar graphs, and 1.62 bits per triangle, or equivalently 3.24 bits per vertex for triangulations. These representations support adjacency queries between vertices and faces in constant time.

A bijection for triangulations of a polygon with interior points and multiple edges

Loopless triangulations of a polygon with k vertices in k + 2n triangles (with interior points and possibly multiple edges) were enumerated by Mullin in 1965, using generating functions and calculations with the quadratic method.In this article we propose a simple bijective interpretation of Mullins formula. The argument rests on the method of conjugacy classes of trees, a variation of the cycle lemma designed for planar maps. In the much easier case of loopless triangulations of the sphere (k = 3), we recover and prove correct an unpublished construction of the second author.