h-graphs: A new representation for tree decompositions of graphs

Abstract In geometric constraint solving, 2D well constrained geometric problems can be abstracted as Laman graphs. If the graph is tree decomposable, the constraint-based geometric problem can be solved by a Decomposition–Recombination planner based solver. In general decomposition and recombination steps can be completed only when steps on which they are dependent have already been completed. This fact naturally defines a hierarchy in the decomposition–recombination steps that traditional tree decomposition representations do not capture explicitly. In this work we introduce h-graphs, a new representation for decompositions of tree decomposable Laman graphs, which captures dependence relations between different tree decomposition steps. We show how h-graphs help in efficiently computing parameter ranges for which solution instances to well constrained, tree decomposable geometric constraint problems with one degree of freedom can actually be constructed.

[1]  Sebastià Vila-Marta,et al.  On the domain of constructive geometric constraint solving techniques , 2001, Proceedings Spring Conference on Computer Graphics.

[2]  Vadim Shapiro,et al.  Boundary representation deformation in parametric solid modeling , 1998, TOGS.

[3]  Jean-Xavier Rampon,et al.  On the calculation of transitive reduction - closure of orders , 1993, Discret. Math..

[4]  Christoph M. Hoffmann,et al.  Decomposition Plans for Geometric Constraint Systems, Part I: Performance Measures for CAD , 2001, J. Symb. Comput..

[5]  Menghan Wang,et al.  Cayley configuration spaces of 2D mechanisms, Part I: extreme points, continuous motion paths and minimal representations , 2011 .

[6]  Robert Joan-Arinyo,et al.  The Reachability Problem in Constructive Geometric Constraint Solving Based Dynamic Geometry , 2013, Journal of Automated Reasoning.

[7]  C. Hoffmann,et al.  A Brief on Constraint Solving , 2005 .

[8]  Vadim Shapiro,et al.  What is a parametric family of solids? , 1995, Symposium on Solid Modeling and Applications.

[9]  Vadim Shapiro,et al.  Necessary conditions for boundary representation variance , 1997, SCG '97.

[10]  H. Garcia,et al.  Geometric constraint solving in a dynamic geometry framework. , 2013 .

[11]  Willem F. Bronsvoort,et al.  A constructive approach to calculate parameter ranges for systems of geometric constraints , 2006, Comput. Aided Des..

[12]  N. Mata,et al.  Applying Constructive Geometric Constraint Solvers to Geometric Problems with Interval Parameters , 2001 .

[13]  Vadim Shapiro,et al.  Consistent updates in dual representation systems , 1999, SMA '99.

[14]  G. Laman On graphs and rigidity of plane skeletal structures , 1970 .

[15]  Jan Willem Klop,et al.  Term Rewriting Systems: From Church-Rosser to Knuth-Bendix and Beyond , 1990, ICALP.

[16]  Alfred V. Aho,et al.  The Transitive Reduction of a Directed Graph , 1972, SIAM J. Comput..

[17]  Christoph M. Hoffmann,et al.  Towards valid parametric CAD models , 2001, Comput. Aided Des..

[18]  Robert Joan-Arinyo,et al.  A constraint-based dynamic geometry system , 2008, SPM '08.

[19]  Christoph M. Hoffmann,et al.  Constraint-based parametric conics for CAD , 1996, Comput. Aided Des..

[20]  Christoph M. Hoffmann,et al.  Decomposition Plans for Geometric Constraint Problems, Part II: New Algorithms , 2001, J. Symb. Comput..

[21]  Robert Joan-Arinyo,et al.  A correct rule-based geometric constraint solver , 1997, Comput. Graph..

[22]  H. A. Van der Meiden Semantics of families of objects , 2008 .

[23]  Robert Joan-Arinyo,et al.  Computing parameter ranges in constructive geometric constraint solving: Implementation and correctness proof , 2012, Comput. Aided Des..

[24]  Menghan Wang,et al.  Cayley Configuration Spaces of 1-dof Tree-decomposable Linkages, Part II: Combinatorial Characterization of Complexity , 2011, ArXiv.

[25]  Gilles Trombettoni,et al.  Decomposition of Geometric Constraint Systems: a Survey , 2006, Int. J. Comput. Geom. Appl..