A task graph with communication costs can be represented as a directed acyclic graph (DAG) $G=(V,E,\omega, c)$.
Here $V = {v_1, ..., v_n}$ represent the set of tasks and the directed edges $E \subseteq V \times V$ represent the
dependencies of the tasks. The weighting functions
$\omega : V \mapsto \mathbb{R}_ {0}^{+}$ and $c : E \mapsto \mathbb{R}_ {0}^{+}$
represent the cost of executing a task and the communication cost between two tasks, respectively.
An object of the BlockIteration class can automatically create the task graph for an explicit setting of the number
of blocks and the number of iterations per block. This graph can be plotted using the plotGraph function. Note that
the procedure described below works completely internally, so the user usually does not interact with the classes.
For each time $n=1,...,N$ and each iteration $k=1...,K$, i.e. specifically $u_{n}^{k}$, a rule is created based on the
block iteration and the initial guess $u_0^0$. This rule is simplified using existing rules and mathematical operations.
The pseudocode for the optimization looks like this:
for all iterations k=1,...,K
for all blocks n=1,...,N
Create rule to calculate $u_{n}^{k}$ based on $u_{0}^{0}$
Simplify the rule based on mathematical operations and existing rules
The rule for a state $u_{n}^{k}$ consists similarly to a block iteration of block operators and previous states
$u_{x}^{y}, x \leq n, y \leq k$. Tasks including their dependencies are derived from these rules and collected in a
`TaskPool. This TaskPool can be easily used to create the graph by iterating through all the tasks and adding a node
and edges for the dependencies for each task.