A Critical Path Approach to Analyzing Parallelism of Algorithmic Variants. Application to Cholesky Inversion

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.