  Algorithms come with multiple variants which are obtained by changing the
mathematical approach from which the algorithm is derived. These variants offer
a wide spectrum of performance when implemented on a multicore platform and we
seek to understand these differences in performances from a theoretical point
of view. To that aim, we derive and present the critical path lengths of each
algorithmic variant for our application problem which enables us to determine a
lower bound on the time to solution. This metric provides an intuitive grasp of
the performance of a variant and we present numerical experiments to validate
the tightness of our lower bounds on practical applications. Our case study is
the Cholesky inversion and its use in computing the inverse of a symmetric
positive definite matrix.
