### Counting and Enumerating Pointed Pseudotriangulations with the Greedy Flip Algorithm

Hervé Brönnimann, Lutz Kettner, Michel Pocchiola, Jack Snoeyink
2006 SIAM journal on computing (Print)
This paper studies pseudo-triangulations for a given point set in the plane. Pseudo-triangulations have many properties of triangulations, and have more freedom since polygons with more than three vertices are allowed as long as they have exactly three inner angles less than π. In particular, there is a natural flip operation on every internal edge. We present an algorithm to enumerate the pseudo-triangulations of a given point set, based on the greedy flip algorithm of Pocchiola and Vegter
more » ... ologically sweeping visibility complexes via pseudo-triangulations; Discrete Comput. Geom. 16:419-453, 1996]. Our two independent implementations agree, and allow us to experimentally verify or disprove conjectures on the numbers of pseudo-triangulations and triangulations of a given point set. (For example, we establish that the number of triangulations is bounded by than the number of pseudo-triangulations for all sets of up to 10 points.) * . 2 8.12n+O(log n) by Denny and Sohler [20]. There are examples of point sets with as many as √ 72 n−Θ(log n) = Ω(8.48 n ) triangulations [8], and it is even known that for any point set S in general position in the plane, #T (S) is Ω(2.33 n ) [9]. Aichholzer [3] has a counting algorithm (that can be executed from a web page for small point sets [2]) and Bespamyatnikh [14] and Ray and Seidel [46] present enumeration algorithms. There remain elementary open questions, such as what point sets have the most and the fewest triangulations. (Aichholzer [2] maintains a list of the leading examples for up to 20 points.) Less is known about the number of pseudo-triangulations, #P T (S), of a given point set S. Even the following conjecture is open: Conjecture 1 [19] For any set S of points in general position in the plane, #T (S) ≤ #P T (S) with equality iff the points are in convex position. The equality when the points are in convex position was recently proven [6]. Randall et al. [45] have established an upper bound #P T (S) ≤ 3 i #T (S) for any point set S with i points inside the convex hull. When combined with the bound on the number of triangulations, this gives #P T (S) ≤ 177.3 n+o(n) . Bespamyatnikh has extended his enumeration algorithm [14] to pseudo-triangulations, but has yet to implement it. Also, his algorithm cannot take a fixed set K of edges and enumerate only the pseudo-triangulations which contain K, which our algorithm can do. This in turn implies a strong connectivity property that is useful when studying the flip-graph (Section 2.2, see also [47] ). Our algorithm, presented in Section 3, is based on the greedy flip algorithm of Pocchiola and Vegter for computing the visibility complex of a scene of n convex objects in the plane [40] . In Section 4, we provide some implementation details; we have produced two independent implementations, which may be obtained from www.cs.poly.edu/pstoolkit/ and www.cs.unc.edu/Research/compgeom/pseudoT/. In Section 5, we present the results of experiments that explore basic conjectures on the number of pseudo-triangulations and triangulations. Both implementations agree in these experiments. Note that Tutte [51] and others have studied the number of topological embeddings of triangulations and rooted triangulations when the locations of vertices are not specified. Li and Nakano [35] enumerate topologically-distinct triangulations with a prescribed number of points on their boundary. We focus strictly on the geometric questions when the vertex set must be a given set of points in the plane. This work was begun at a Bellairs workshop on pseudo-triangulations organized by Ileana Streinu and partially supported by the NSF. The published results by the participants include the numbers of pseudo-triangulations of special point configurations [45] , the existence of pseudo-triangulations with bounded degree [27, 28] , and an analysis of the flip graph [47] . Especially this last work quotes some of the results on flipping contained in this paper.