Exploiting Common Subexpressions in Numerical CSPs

  • Ignacio Araya
  • Bertrand Neveu
  • Gilles Trombettoni
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5202)


It is acknowledged that the symbolic form of the equations is crucial for interval-based solving techniques to efficiently handle systems of equations over the reals. However, only a few automatic transformations of the system have been proposed so far. Vu, Schichl, Sam-Haroud, Neumaier have exploited common subexpressions by transforming the equation system into a unique directed acyclic graph. They claim that the impact of common subexpressions elimination on the gain in CPU time would be only due to a reduction in the number of operations.

This paper brings two main contributions. First, we prove theoretically and experimentally that, due to interval arithmetics, exploiting certain common subexpressions might also bring additional filtering/contraction during propagation. Second, based on a better exploitation of n-ary plus and times operators, we propose a new algorithm I-CSE that identifies and exploits all the “useful” common subexpressions. We show on a sample of benchmarks that I-CSE detects more useful common subexpressions than traditional approaches and leads generally to significant gains in performance, of sometimes several orders of magnitude.


Auxiliary Variable Equality Node Interval Analysis Interval Arithmetic Narrowing Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Benhamou, F., Goualard, F., Granvilliers, L., Puget, J.-F.: Revising Hull and Box Consistency. In: Proc. ICLP, pp. 230–244 (1999)Google Scholar
  2. 2.
    Brown, D.P.: Calculus and Mathematica. Addison Wesley, Reading (1991)Google Scholar
  3. 3.
    Buchberger, B.: Gröbner Bases: an Algorithmic Method in Polynomial Ideal Theory. Multidimensional Systems Theory, 184–232 (1985)Google Scholar
  4. 4.
    Ceberio, M., Granvilliers, L.: Solving Nonlinear Equations by Abstraction, Gaussian Elimination, and Interval Methods. In: Armando, A. (ed.) FroCos 2002. LNCS (LNAI), vol. 2309. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  5. 5.
    Chabert, G. (2008),
  6. 6.
    Collavizza, H., Delobel, F., Rueher, M.: Comparing partial consistencies. Reliable Computing 5(3), 213–228 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Debruyne, R., Bessière, C.: Some Practicable Filtering Techniques for the Constraint Satisfaction Problem. In: Proc. IJCAI, pp. 412–417 (1997)Google Scholar
  8. 8.
    Flajolet, P., Sipala, P., Steyaert, J.-M.: Analytic variations on the common subexpression problem. In: Paterson, M. (ed.) ICALP 1990. LNCS, vol. 443, pp. 220–334. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  9. 9.
    Granvilliers, L., Monfroy, E., Benhamou, F.: Symbolic-Interval Cooperation in Constraint Programming. In: Proc. ISSAC, pp. 150–166. ACM, New York (2001)CrossRefGoogle Scholar
  10. 10.
    Harvey, W., Stuckey, P.J.: Improving Linear Constraint Propagation by Changing Constraint Representation. Constraints 7, 173–207 (2003)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Heck, A.: Introduction to Maple. Springer, Heidelberg (2003)zbMATHGoogle Scholar
  12. 12.
    Lebbah, Y.: Contribution à la Résolution de Contraintes par Consistance Forte. Phd thesis, Université de Nantes (1999)Google Scholar
  13. 13.
    Lhomme, O.: Consistency Tech. for Numeric CSPs. In: IJCAI, pp. 232–238 (1993)Google Scholar
  14. 14.
    Merlet, J.-P.: ALIAS: An Algorithms Library for Interval Analysis for Equation Systems. Technical report, INRIA Sophia (2000),
  15. 15.
    Merlet, J.-P.: Interval Analysis and Robotics. In: Symp. of Robotics Research (2007)Google Scholar
  16. 16.
    Muchnick, S.: Advanced Compiler Design and Implem. M. Kauffmann (1997)Google Scholar
  17. 17.
    Neumaier, A.: Interval Methods for Systems of Equations. Cambridge University Press, Cambridge (1990)zbMATHGoogle Scholar
  18. 18.
    Schichl, H., Neumaier, A.: Interval analysis on directed acyclic graphs for global optimization. Journal of Global Optimization 33(4), 541–562 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Trombettoni, G., Chabert, G.: Constructive Interval Disjunction. In: Bessière, C. (ed.) CP 2007. LNCS, vol. 4741, pp. 635–650. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  20. 20.
    Vu, X.-H., Schichl, H., Sam-Haroud, D.: Using Directed Acyclic Graphs to Coordinate Propagation and Search for Numerical Constraint Satisfaction Problems. In: Proc. ICTAI 2004, pp. 72–81. IEEE, Los Alamitos (2004)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Ignacio Araya
    • 1
  • Bertrand Neveu
    • 1
  • Gilles Trombettoni
    • 1
  1. 1.INRIAUniversité de Nice-SophiaCertis

Personalised recommendations