Improvements to Buchberger's Algorithm generally seek either to define a criterion for the removal of unnecessary S-pairs or to describe a strategy for improving the choices which one must make in the course of the algorithm. This paper surveys significant improvements to Buchberger's original algorithm for Groebner basis computation including the Gebauer-Moeller Criteria, the "Sugar" strategy, and Jean-Charles Faugere's F4 algorithm. Since Faugere's F4 is generally accepted as being a particularly efficient approach to Groebner basis computation, we test several variants of the F4 algorithm on a variety of benchmark ideals in an effort to judge the efficiency of the Groebner basis computation process, while also being mindful of the memory constraint issues occurring in computer algebra.