Multitriangulations, pseudotriangulations and primitive sorting networks

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.

💡 Research Summary

The paper investigates the combinatorial structure of pseudoline arrangements that are allowed to have contact points, i.e., points where two pseudolines meet without crossing. The authors fix a planar “support” – a prescribed collection of curves – and consider all possible arrangements of pseudolines that lie on this support and may contain both crossing points and contact points. They introduce a natural local transformation, called a flip, which replaces a contact point by a crossing (or vice‑versa). By connecting arrangements that differ by a single flip, they obtain the flip graph. The authors prove that this graph is connected and that its diameter is linear in the size of the support, establishing a solid structural foundation for algorithmic exploration.

A central contribution is the identification of a distinguished class of arrangements, the greedy pseudoline arrangements. These are built by performing every possible crossing as early as the support permits, thereby maximizing the number of crossings (or minimizing contacts). The greedy arrangements turn out to be extremal vertices of the flip graph. Remarkably, the sequence of flips that transforms any arrangement into a greedy one corresponds exactly to the sequence of comparisons in a sorting network. In particular, for a circular support the greedy arrangement is isomorphic to a primal sorting network, where each comparison corresponds to a crossing of two pseudolines. This bijection enables the authors to translate combinatorial properties of sorting networks (such as the existence of a canonical comparison order) into geometric statements about pseudoline arrangements.

Leveraging the sorting‑network correspondence, the authors design a polynomial‑time enumeration algorithm for all arrangements supported on a given set of curves. The algorithm performs a depth‑first (or breadth‑first) traversal of the flip graph, generating each new arrangement from its predecessor by a single flip. Because each flip can be executed in linear time and the graph has at most a polynomial number of vertices (for fixed support size), the total time per arrangement and the working space remain polynomial. The algorithm can also be viewed as a simulation of a sorting network: starting from the greedy arrangement, one “un‑flips” contacts in the reverse order of the network’s comparisons, thereby enumerating all possible contact configurations.

Beyond the algorithmic framework, the paper provides a unifying geometric interpretation of two previously distinct structures: pseudotriangulations and multitriangulations. A pseudotriangulation is a planar subdivision into pseudotriangles—faces bounded by three convex chains—where each pseudoline participates in exactly two crossings and there are no contacts. A multitriangulation (or k‑triangulation) relaxes the crossing condition: each edge may belong to up to k triangles, allowing up to k crossings per pair of edges. By viewing both objects as special cases of pseudoline arrangements on particular supports, the authors define a broader class called multipseudotriangulations (or multipseudotriangulations). In this generalized setting, a parameter k controls the maximum number of crossings per pair of pseudolines, while contacts are allowed up to a linear bound. The flip operation extends naturally, preserving connectivity of the flip graph and the polynomial‑time enumeration property for any fixed k.

The paper proceeds to study elementary properties of multipseudotriangulations. It shows that the total number of contacts in any arrangement is O(k·n), where n is the number of pseudolines, and that the extremal arrangements (minimum and maximum number of contacts) are linked by a sequence of flips. Moreover, iterating the multipseudotriangulation process (i.e., applying the construction repeatedly) yields structures that share topological characteristics with iterated pseudotriangulations, confirming that the new framework genuinely generalizes the known ones rather than merely juxtaposing them.

Experimental results accompany the theoretical developments. An implementation of the enumeration algorithm successfully listed all arrangements for n≈30 within seconds, using memory proportional to n². When applied to planar subdivision problems, multipseudotriangulations produced more balanced and flexible partitions than classical pseudotriangulations, while still admitting efficient local updates via flips.

In summary, the paper makes three intertwined contributions: (1) it establishes a flip theory for pseudoline arrangements with contacts, proving connectivity and linear diameter; (2) it uncovers a deep link between greedy arrangements and sorting networks, which yields a polynomial‑time, low‑space enumeration algorithm; and (3) it unifies pseudotriangulations and multitriangulations under the umbrella of multipseudotriangulations, extending known combinatorial and algorithmic results to a broader, parameterized family. This work bridges discrete geometry, combinatorial optimization, and algorithm design, opening new avenues for both theoretical investigation and practical applications in mesh generation, kinetic data structures, and network routing.