Michel Pocchiola

Topologically sweeping visibility complexes via pseudotriangulations

This paper describes a new algorithm for constructing the set of free bitangents of a collection ofn disjoint convex obstacles of constant complexity. The algorithm runs in timeO(n logn + k), where,k is the output size, and uses,O(n) space. While earlier algorithms achieve the same optimal running time, this is the first optimal algorithm that uses only linear space. The visibility graph or the visibility complex can be computed in the same time and space. The only complicated data structure used by the algorithm is a splittable queue, which can be implemented easily using red-black trees. The algorithm is conceptually very simple, and should therefore be easy to implement and quite fast in practice. The algorithm relies on greedy pseudotriangulations, which are subgraphs of the visibility graph with many nice combinatorial properties. These properties, and thus the correctness of the algorithm, are partially derived from properties of a certain partial order on the faces of the visibility complex.

Computing a canonical polygonal schema of an orientable triangulated surface

A closed orientable surface of genus

Pseudo-triangulations: theory and applications

g

Computing the visibility graph via pseudo-triangulations

can be obtained by appropriat e identification of pairs of edges of a

Minimal tangent visibility graphs

4g

A GEOMETRIC PROOF OF THE ENUMERATION FORMULA FOR STURMIAN WORDS

-gon (the polygonal schema). The identified edges form

Counting and Enumerating Pointed Pseudotriangulations with the Greedy Flip Algorithm

2g

Average cost of Duval's algorithm for generating Lyndon words

loops on the surface, that are disjoint except for their common end-point. These loops are generators of both the fundamental group and the homology group of the surface. The inverse problem is concerned with finding a set of

Random generation of finite Sturmian words

2g

Tutte's barycenter method applied to isotopies

loops on a triangulated surface, such that cutting the surface along these loops yields a (canonical) polygonal schema. We present two optimal algorithms for this inverse problem. Both algorithms have been implemented using the CGAL polyhedron data structure.