Polynomial systems are ubiquitous in mathematics, science and engineering. Gr?bner basis theory is one of the most powerful tools for solving polynomial systems and is essential in many computational tasks in algebra and algebraic geometry. Buchberger introduced in ???? the concept of Gr?bner basis and the first algorithm for computing Gr?bner bases, and this algorithm has been implemented in most computer algebra systems including Maple, Mathematica, Magma, Sage, Singular, Macaulay ?, CoCoA, etc. There has been extensive effort in finding more efficient algorithms for computing Gr?bner bases. In Buchberger’s original algorithm, one has to reduce many useless S-polynomials (i.e., those that reduce to ? via long division), and each reduction is time consuming.