Multitriangulations, pseudotriangulations and primitive sorting networks

TL;DR

This paper explores the structure of pseudoline arrangements and introduces algorithms for enumeration, providing new insights into pseudotriangulations and multitriangulations through a novel geometric interpretation.

Contribution

It offers a new geometric interpretation of pseudotriangulations and multitriangulations via pseudoline arrangements, and develops a polynomial-time enumeration algorithm for arrangements with a given support.

Findings

Enumeration algorithm with polynomial time and space complexity

New interpretation linking pseudotriangulations and multitriangulations

Introduction of multipseudotriangulations as a generalization

Abstract

We study the set of all pseudoline arrangements with contact points which cover a given support. We define a natural notion of flip between these arrangements and study the graph of these flips. In particular, we provide an enumeration algorithm for arrangements with a given support, based on the properties of certain greedy pseudoline arrangements and on their connection with sorting networks. Both the running time per arrangement and the working space of our algorithm are polynomial. As the motivation for this work, we provide in this paper a new interpretation of both pseudotriangulations and multitriangulations in terms of pseudoline arrangements on specific supports. This interpretation explains their common properties and leads to a natural definition of multipseudotriangulations, which generalizes both. We study elementary properties of multipseudotriangulations and compare them to iterations of pseudotriangulations.