Upper and Lower Bounds on the Makespan of Schedules for Tree Dags on Linear Arrays

Abstract.

We find, in polynomial time, a schedule for a complete binary tree directed acyclic graph (dag) with n unit execution time tasks on a linear array whose makespan is optimal within a factor of 1+o(1) . Further, given a binary tree dag T with n tasks and height h , we find, in polynomial time, a schedule for T on a linear array whose makespan is optimal within a factor of 5 + o(1) .

On the other hand, we prove that explicit lower and upper bounds on the makespan of optimal schedules of binary tree dags on linear arrays differ at least by a factor of 1+ $\sqrt{2}/2$ . We also find, in polynomial time, schedules for bounded tree dags with n unit execution time tasks, degree d , and height $h \in o(n^{1/2}) \cup \omega(n^{1/2})$ on a linear array which are optimal within a factor of 1+o(1) , this time under the assumption of links with unlimited bandwidth.

Finally, we compute an improved upper bound on the makespan of an optimal schedule for a tree dag on the architecture independent model of Papadimitriou and Yannakakis [14], provided that its height is not too large.