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| <a name="Omega-a-solver-for-linear-programming-problems"></a> |
| <h3 class="section">15.9 Omega a solver for linear programming problems</h3> |
| <a name="index-Omega-a-solver-for-linear-programming-problems"></a> |
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| <p>The data dependence analysis contains several solvers triggered |
| sequentially from the less complex ones to the more sophisticated. |
| For ensuring the consistency of the results of these solvers, a data |
| dependence check pass has been implemented based on two different |
| solvers. The second method that has been integrated to GCC is based |
| on the Omega dependence solver, written in the 1990’s by William Pugh |
| and David Wonnacott. Data dependence tests can be formulated using a |
| subset of the Presburger arithmetics that can be translated to linear |
| constraint systems. These linear constraint systems can then be |
| solved using the Omega solver. |
| </p> |
| <p>The Omega solver is using Fourier-Motzkin’s algorithm for variable |
| elimination: a linear constraint system containing <code>n</code> variables |
| is reduced to a linear constraint system with <code>n-1</code> variables. |
| The Omega solver can also be used for solving other problems that can |
| be expressed under the form of a system of linear equalities and |
| inequalities. The Omega solver is known to have an exponential worst |
| case, also known under the name of “omega nightmare” in the |
| literature, but in practice, the omega test is known to be efficient |
| for the common data dependence tests. |
| </p> |
| <p>The interface used by the Omega solver for describing the linear |
| programming problems is described in <samp>omega.h</samp>, and the solver is |
| <code>omega_solve_problem</code>. |
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