Memory needed for computational graph Suppose we have a set of equations like this

p7=f(p1+p6); p6=f(p2+p5); p5=f(p3+p4); p4=f(p3); p3=f(p2); p2=f(p1); p1=f()


It can be represented by computational graph below

 

If each intermediate value takes 1 unit of memory, you need at least 4 units to compute p7 without any duplicate computation.

Is there an algorithm for estimating memory needed in this setting for a general DAG?

I found a paper called "Adjoint Dataflow Analysis" for estimating this for restricted set of graphs, but it feels like this ought to be a problem that is covered more generally in graph theory.
Your problem sounds similar to one-shot (black) pebbling. Wu, Austrin, Pitassi, and Liu, in their paper titled Inapproximability of treewidth, one-shot pebbling, and related layout problems (J. Artificial Intelligence Res. 49 (2014), 569–600), show that it is (probably) hard to compute the optimal cost (which corresponds to your memory).