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GA and CHC. Two Evolutionary Algorithms to Solve the Root Identification Problem in Geometric Constraint Solving

  • M. V. Luzón
  • E. Barreiro
  • E. Yeguas
  • R. Joan-Arinyo
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3039)

Abstract

Geometric problems defined by constraints have an exponential number of solution instances in the number of geometric elements involved. Generally, the user is only interested in one instance such that, besides fulfilling the geometric constraints, exhibits some additional properties.

Selecting a solution instance amounts to selecting a given root every time the geometric constraint solver needs to compute the zeros of a multi valuated function. The problem of selecting a given root is known as the Root Identification Problem.

In this paper we present a comparative study of a basic genetic algorithm against the CHC algorithm. Both techniques are based on an automatic search in the space of solutions driven by a set of extra constraints. A number of case studies illustrate the performance of the methods.

Keywords

Evolutionary algorithms Constructive geometric constraint solving Root identification problem Solution selection 

References

  1. 1.
    Bouma, W., Fudos, I., Hoffmann, C., Cai, J., Paige, R.: Geometric constraint solver. Computer-Aided Design 27(6), 487–501 (1995)zbMATHCrossRefGoogle Scholar
  2. 2.
    Bremermann, H.J., Roghson, J., Salaff, S.: Global properties of evolution processes. In: Pattee, H.H., Edelsack, E.A., Fein, L., Callahan, A.B. (eds.) Natural Automata and Useful Simulations, pp. 3–42. Macmillan, Basingstoke (1966)Google Scholar
  3. 3.
    Brüderlin, B.D.: Rule-Based Geometric Modelling. PhD thesis, Institut für Informatik der ETH Zürich (1988)Google Scholar
  4. 4.
    Durand, C.: Symbolic and Numerical Techniques for Constraint Solving. PhD thesis, Purdue University, Department of Computer Sciences (December 1998)Google Scholar
  5. 5.
    Eiben, A.E., Ruttkay, Z.: Constraint-satisfaction problems. In: Bäck, T., Fogel, D.B., Michalewicz, Z. (eds.) Handbook of Evolutionary Computation. ch. C5.7, pp. C5.7:1–C5.7:5. Institute of Physics Publishing Ltd and Oxford University Press (1997)Google Scholar
  6. 6.
    Eshelman, L.J.: The CHC adaptive search algorithm: How to safe search when engaging in nontraditional genetic recombination. In: Foundations of Genetic Algorithms, pp. 265–283 (1991)Google Scholar
  7. 7.
    Essert-Villard, C., Schreck, P., Dufourd, J.-F.: Sketch-based pruning of a solution space within a formal geometric constraint solver. Artificial Intelligence 124, 139–159 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Forrest, S. (ed.): Proceedings of the Fifth International Conference on Genetic Algorithms, San Mateo, CA. Morgan Kaufmann, San Francisco (1993)Google Scholar
  9. 9.
    Fudos, I., Hoffmann, C.M.: Correctness proof of a geometric constraint solver. International Journal of Computational Geometry & Applications 6(4), 405–420 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Fudos, I., Hoffmann, C.M.: A graph-constructive approach to solving systems of geometric constraints. ACM Transactions on Graphics 16(2), 179–216 (1997)CrossRefGoogle Scholar
  11. 11.
    Goldberg, D.E.: Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, Reading (1989)zbMATHGoogle Scholar
  12. 12.
    Holland, J.H.: Adaptation in Natural and Artificial Systems. The University of Michigan Press, Ann Arbor (1975)Google Scholar
  13. 13.
    Joan-Arinyo, R., Soto-Riera, A.: Combining geometric and equational geometric constraint solving techniques. In: VII Congreso Español de Informática Gráfica, Eurographics, June 1997, pp. 309–324 (1997)Google Scholar
  14. 14.
    Joan-Arinyo, R., Soto-Riera, A.: Combining constructive and equational geometric constraint solving techniques. ACM Transactions on Graphics 18(1), 35–55 (1999)CrossRefGoogle Scholar
  15. 15.
    Joan-Arinyo, R., Luzón, M.V., Soto, A.: Constructive geometric constraint solving: a new application of genetic algorithms. In: Parallel Problem Solving from Nature-PPSN VII, vol. 1, pp. 759–768 (2002)Google Scholar
  16. 16.
    Joan-Arinyo, R., Luzón, M.V., Soto, A.: Genetic algorithms for root multiselection in constructive geometric constraint solving. Computer & Graphics 27, 51–60 (2003)CrossRefGoogle Scholar
  17. 17.
    Kleene, S.C.: Mathematical Logic. John Wiley and Sons, New York (1967)zbMATHGoogle Scholar
  18. 18.
    Laman, G.: On graphs and rigidity of plane skeletal structures. Journal of Engineering Mathematics 4(4), 331–340 (1970)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Lovász, L., Yemini, Y.: On generic rigidity in the plane. SIAM Journal on Algebraic and Discrete Methods 3(1), 91–98 (1982)zbMATHCrossRefGoogle Scholar
  20. 20.
    Luzón, M.V.: Resolución de Restricciones geométricas. Selección de la solución deseada. PhD thesis, Dpto. de Informática. Universidade de Vigo. (September 2001)Google Scholar
  21. 21.
    Mata, N.: Solving incidence and tangency constraints in 2D. Technical Report LSI-97-3R, Department LiSI, Universitat Politècnica de Catalunya (1997)Google Scholar
  22. 22.
    Mata, N.: Constructible Geometric Problems with Interval Parameters. PhD thesis, Dept. LSI, Universitat Politècnica de Catalunya, Barcelona, Catalonia, Spain (2000)Google Scholar
  23. 23.
    Mendenhall, W., Sincich, T.: Statistics for engineering and the sciences, 4th edn. Prentice-Hall, Englewood Cliffs (1999)Google Scholar
  24. 24.
    Michalewicz, Z.: Genetic Algorithms + Data Structures = Evolution Programs. Springer, Heidelberg (1996)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • M. V. Luzón
    • 1
  • E. Barreiro
    • 1
  • E. Yeguas
    • 1
  • R. Joan-Arinyo
    • 2
  1. 1.Escuela Superior de Ingeniería InformáticaUniversidade de VigoOurense
  2. 2.Escola Técnica Superior d’Enginyeria Industrial de BarcelonaUniversitat Politècnica de CatalunyaBarcelona

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