Revisiting decomposition analysis of geometric constraint graphs

Abstract

Geometric problems defined by constraints can be represented by geometric constraint graphs whose nodes are geometric elements and whose arcs represent geometric constraints. Reduction and decomposition are techniques commonly used to analyze geometric constraint graphs in geometric constraint solving.In this paper we first introduce the concept of deficit of a constraint graph. Then we give a new formalization of the decomposition algorithm due to Owen. This new formalization is based on preserving the deficit rather than on computing triconnected components of the graph and is simpler. Finally we apply tree decompositions to prove that the class of problems solved by the formalizations studied here and other formalizations reported in the literature is the same.

Geometric Constraint Solving Based on Connectivity of Graph

“…This conclusion is the same as that in [16,28], based on which Owen gave an efficient algorithm and Joan-Arinyo et al gave an improved algorithm in 2D.…”

“…to be structurally well-constrained and its actual number of weight |E| in 2D [16]. Here we generalize the deficit function to more general cases.…”

“…There are four major approaches to geometric constraint solving: the numerical approach [12,21,27], the symbolic computation approach [7,19], the rule-based approach [2,6,20,29] and the graph-based approach [3,4,14,16,23,24,26]. This paper will focus on using graph algorithms to decompose a large constraint problem into smaller ones.…”

“…In [5,13], Hoffmann et al proposed a method based on cluster formation to solve 2D and 3D constraint problems. In [16], Joan-Arinyo et al proposed an algorithm to decompose a 2D constraint problem into an s-tree. This method is equivalent to Owen's and Hoffmann's methods, but is conceptually simpler.…”

“…D-tree). This method is a natural generalization of the methods in [28,16] which are based on tri-connectivity analysis of the constrained graph, and can solve problems that can be reduced to triangles. On the other hand, our method can be used to deal with general problems.…”