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

Abstract

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 edgeconstrained 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.

Fast Enumeration Algorithms for Non-crossing Geometric Graphs

“…In this section, we first introduce some notation used throughout the paper and then provide a number of preliminary results on the F -constrained lexicographically largest triangulation (F -CLLT). These results were originally obtained in [21] for the purpose of developing an efficient enumeration algorithm for edgeconstrained non-crossing spanning trees and play a crucial role in the development of our framework.…”

“…For some specific graph classes, Avis et al [9] and the authors [21] have shown how to avoid duplicate generation based on a well-known enumeration framework, called the reverse search [6,7]. In this paper we extend this idea and develop a new general technique which does not rely on the property of a particular graph class.…”

“…Moreover, from the viewpoint of the applications to enumeration problems, a transformation should be defined locally, i.e., the symmetric difference between two connected objects should be as small as possible. Developing such a nice transformation might be interesting in its own right, and there are many known results not only for local transformations [2,5,7,[21][22][23]25] but also for large transformations [1,3,24]. Almost all previous works for the enumerations of non-crossing geometric graphs discussed above are based on local transformations, and thereby they deeply rely on the property of a particular graph class.…”

“…Enumerating combinatorial objects is a fundamental problem, and several algorithms have been developed for non-crossing geometric graphs, e.g., triangulations [7,12], non-crossing spanning trees [2,7,21], pseudo-triangulations [10,13], and noncrossing minimally rigid graphs [8,9]. Let us explain why the enumeration of noncrossing geometric graphs is more difficult than that of non-geometric (abstract) graphs.…”

“…In this paper we extend this idea and develop a new general technique which does not rely on the property of a particular graph class. In order to avoid duplicate enumeration, we introduce two key notions: edgeconstrained lexicographically largest triangulations (which were originally introduced in [21] for the development of an efficient enumeration algorithm of edgeconstrained non-crossing spanning trees) and minimal representative sets. For a set of non-crossing segments F , a geometric graph containing F is called F -constrained.…”