Exploiting Common Subexpressions in Numerical CSPs
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.
KeywordsAuxiliary Variable Equality Node Interval Analysis Interval Arithmetic Narrowing Operator
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- 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.Brown, D.P.: Calculus and Mathematica. Addison Wesley, Reading (1991)Google Scholar
- 3.Buchberger, B.: Gröbner Bases: an Algorithmic Method in Polynomial Ideal Theory. Multidimensional Systems Theory, 184–232 (1985)Google Scholar
- 5.Chabert, G. (2008), http://ibex-lib.org
- 7.Debruyne, R., Bessière, C.: Some Practicable Filtering Techniques for the Constraint Satisfaction Problem. In: Proc. IJCAI, pp. 412–417 (1997)Google Scholar
- 12.Lebbah, Y.: Contribution à la Résolution de Contraintes par Consistance Forte. Phd thesis, Université de Nantes (1999)Google Scholar
- 13.Lhomme, O.: Consistency Tech. for Numeric CSPs. In: IJCAI, pp. 232–238 (1993)Google Scholar
- 14.Merlet, J.-P.: ALIAS: An Algorithms Library for Interval Analysis for Equation Systems. Technical report, INRIA Sophia (2000), http://www-sop.inria.fr/coprin/logiciels/ALIAS/ALIAS.html
- 15.Merlet, J.-P.: Interval Analysis and Robotics. In: Symp. of Robotics Research (2007)Google Scholar
- 16.Muchnick, S.: Advanced Compiler Design and Implem. M. Kauffmann (1997)Google Scholar
- 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