Enumerating edge-constrained triangulations and edge-constrained non-crossing geometric spanning trees

• Published in: Discrete Applied Mathematics Vol. 157; no. 17; pp. 3569 - 3585
• Main Authors: Katoh, Naoki, Tanigawa, Shin-ichi
• Format: Journal Article Conference Proceeding
• Language: English
• Published: Kidlington Elsevier B.V 28.10.2009; Elsevier
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Combinatorics; Combinatorics. Ordered structures; Computer science; control theory; systems; Edge-constrained non-crossing spanning trees; Edge-constrained triangulations; Exact sciences and technology; Geometric enumeration; Graph theory; Information retrieval. Graph; Mathematics; Sciences and techniques of general use; Theoretical computing; Geometric enumeration; Edge-constrained triangulations; Edge-constrained non-crossing spanning trees; Triangulation; Enumeration; Inclusion; Computer theory; Constraint; Plane; Algorithm; trees; Optimization; Edge-constrained non-crossing spanning; Spanning tree; First order; Ordering; Combinatorics; Edge set; Repetition
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2009.04.019

In this paper, we present algorithms for enumerating, without repetitions, all triangulations and non-crossing geometric spanning trees on a given set of n points in the plane under edge inclusion constraint (i.e., some edges are required to be included in the graph). We will first extend the lexicographically ordered triangulations introduced by Bespamyatnikh to the edge-constrained case, and then we prove that a set of all edge-constrained non-crossing spanning trees is connected via remove-add flips, based on the edge-constrained lexicographically largest triangulation. More specifically, we prove that all edge-constrained triangulations can be transformed to the lexicographically largest triangulation among them by O ( n 2 ) greedy flips, i.e., by greedily increasing the lexicographical ordering of the edge list, and a similar result also holds for a set of edge-constrained non-crossing spanning trees. Our enumeration algorithms generate each output triangulation and non-crossing spanning tree in O ( log log n ) and O ( n 2 ) time, respectively, based on the reverse search technique.