A Makespan Lower Bound for the Scheduling of the Tiled Cholesky Factorization based on ALAP scheduling

Due to the advent of multicore architectures and massive parallelism, the tiled Cholesky factorization algorithm has recently received plenty of attention and is often referenced by practitioners as a case study. It is also implemented in mainstream dense linear algebra libraries. However, we note that theoretical study of the parallelism of this algorithm is currently lacking. In this paper, we present new theoretical results about the tiled Cholesky factorization in the context of a parallel homogeneous model without communication costs. We use standard flop-based weights for the tasks. For a $t$-by-$t$ matrix, we know that the critical path of the tiled Cholesky algorithm is $9t-10$ and that the weight of all tasks is $t^3$. In this context, we prove that no schedule with less than $0.185 t^2$ processing units can finish in a time less than the critical path. In perspective, a naive bound gives $0.11 t^2.$ We then give a schedule which needs less than $0.25 t^2+0.16t+3$ processing units to complete in the time of the critical path. In perspective, a naive schedule gives $0.50 t^2.$ In addition, given a fixed number of processing units, $p$, we give a lower bound on the execution time as follows: $$\max( \frac{t^{3}}{p}, \frac{t^{3}}{p} - 3\frac{t^2}{p} + 6\sqrt{2p} - 7 , 9t-10).$$ The interest of the latter formula lies in the middle term. Our results stem from the observation that the tiled Cholesky factorization is much better behaved when we schedule it with an ALAP (As Late As Possible) heuristic than an ASAP (As Soon As Possible) heuristic. We also provide scheduling heuristics which match closely the lower bound on execution time. We believe that our theoretical results will help practical scheduling studies. Indeed, our results enable to better characterize the quality of a practical schedule with respect to an optimal schedule.