$$w_0^{(eq)} = \left( \sum_i (w'_i)^3 \right)^{\frac{1}{3}} + w_0.$$
Finally the tree is equivalent to one task of cost $w_0^{(eq)}$, and if $\frac{w_0^{(eq)}}{D} \le s_{max}$, the energy consumption is $\frac{(w_0^{(eq)})^3}{D^2}$, and no speed exceeds $s_{max}$.
Note that the speed of a task is always greater than the speed of its successors. Therefore, if $\frac{w_0^{(eq)}}{D} > s_{max}$, we execute the root of the tree at speed $s_{max}$ and then process each subtree $G_i$ independently. Of course, there is no solution if $\frac{w_0}{s_{max}} > D$, and otherwise we perform the recursive calls to tree to process each subtree independently. Their deadline is then $D - \frac{w_0}{s_{max}}$. $\square$
4.2.5. Series-parallel graphs
We can further generalize our results to series-parallel graphs (SPGs), which are built from a sequence of compositions (parallel or series) of smaller-size SPGs. The smallest SPG consists of two nodes connected by an edge (such a graph is called an elementary SPG). The first node is the source, while the second one is the sink of the SPG. When composing two SGPs in series, we merge the sink of the first SPG with the source of the second one. For a parallel composition, the two sources are merged, as well as the two sinks, as illustrated in Figure 2.
We can extend the results for tree graphs to SPGs, by replacing step by step the SPGs by an equivalent task (procedure cost in Algorithm 2): we can compute the equivalent cost for a series or parallel composition.
However, since it is no longer true that the speed of a task is always larger than the speed of its successor (as was the case in a tree), we have not been able to find a recursive property on the tasks that should be set to $s_{max}$, when one of the speeds obtained with the previous method exceeds $s_{max}$. The problem of computing a closed form for a SPG with a finite value of $s_{max}$ remains open. Still, we have the following result when $s_{max} = +\infty$:
Theorem 3 (series-parallel graphs) When G is a SPG, it is possible to compute recursively a closed form expression of the optimal solution of MINENERGY(G, D), assuming $s_{max} = +\infty$, in polynomial time $O(|V|)$, where V is the set of tasks.
Proof. Let $G$ be a series-parallel graph. The optimal solution to $\text{MINENERGY}(G, D)$ is obtained with a call to SPG $(G, D)$, and we prove its optimality recursively. Similarly to trees, the main idea is to peel the graph off, and to transform it until there remains only a single equivalent task that, if executed alone within a deadline $D$, would consume exactly