Scientific workflows are frequently modeled as Directed Acyclic Graphs
(DAG) of tasks, which represent computational modules and their
dependencies in the form of data produced by a task and used by
another one. This formulation allows the use of runtime systems which
dynamically allocate tasks onto the resources of increasingly complex
computing platforms. However, for some workflows, such a dynamic
schedule may run out of memory by exposing too much parallelism. This
paper focuses on the problem of transforming such a DAG to prevent
memory shortage, and concentrates on shared memory platforms. We first
propose a simple model of DAGs which is expressive enough to emulate
complex memory behaviors. We exhibit a polynomial-time algorithm
that computes the maximum peak memory of a DAG, that is, the maximum
memory needed by any parallel schedule. We then consider the problem of
reducing this maximum peak memory to make it smaller than a given
bound by adding new fictitious edges, while trying to minimize the
critical path of the graph.