Fast enumeration algorithms for non-crossing geometric graphs

Abstract

A non-crossing geometric graph is a graph embedded on a given set of points in the plane with non-crossing straight line segments. In this paper we present a new general framework for enumerating non-crossing geometric graphs for a given point set. By applying our idea to specific enumeration problems, we obtain faster algorithms for enumerating plane straight-line graphs, non-crossing spanning connected graphs, non-crossing spanning trees and non-crossing minimally rigid frameworks. Furthermore, we also obtain efficient enumeration algorithms for non-crossing geometric graph classes, for which no enumeration algorithm has been reported so far, such as non-crossing matchings, noncrossing blue-and-red matchings, non-crossing k-vertex or k-edge connected graphs or non-crossing directed spanning trees. The proposed idea is relatively simple, and can be potentially applied to various other enumeration problems of non-crossing geometric graphs.

A simple aggregative algorithm for counting triangulations of planar point sets and related problems

“…There has also been a fair number of papers on computing tr(S) given S: The reverse search method of Avis and Fukuda [7] can be applied to enumerate (see [8] for a particularly fast realization), Katoh and Tanigawa [15] consider more general enumeration problems, Aichholzer [1] proposed a divide-and-conquer method base on so-called triangulation paths, Ray and Seidel [19] exploited dynamic programming, and Alvarez, Bringmann, and Ray [6] applied a sweep approach based on triangulation paths and also proposed a method that exploited the onion layer structure of a point set, see [5]. This last approach achieved the so far best worst case running time with a bound of O(3.1414 n ).…”

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

“…We have also provided several geometric properties of the edge-constrained lexicographically largest triangulation in Sections 2 and 3. In our recent paper [22], using the edge-constrained lexicographically largest triangulation as well as the results of Section 3, we have newly revealed combinatorial properties that relate the non-crossing geometric graphs and the edge-constrained lexicographically largest triangulation on a point set. Based on the properties, we have proposed a general framework for efficiently enumerating a large class of non-crossing geometric graphs such as plane straight-line graphs, non-crossing spanning connected graphs, (unconstrained) non-crossing spanning trees, non-crossing minimally rigid graphs, non-crossing matchings, non-crossing blue-and-red matchings and etc.…”

Counting and Enumerating Crossing-free Geometric Graphs

“…The defined classes of geometric graphs have also been studied from an algorithmic point of view. The problem of enumeration has been solved for G(P ), T (P ) and ST (P ), see [7,8,2,15]. By solved we mean that these sets can be enumerated in such a way that the time delay for each enumerated object is bounded by a polynomial in n. 1 Apply the lower bound for perfect matchings on all subsets of P and then use the binomial theorem.…”