Low-Memory Tour Reversal in Directed Graphs

We consider the problem of reversing a tour (i1, i2, . . . , il) in a directed graph G = (V, E) with positive integer vertices V and edges E ⊆ V × V , where ij ∈ V and (ij, ij+1) ∈ E for all j = 1, . . . , l − 1. The tour can be processed in last-in-first-out order as long as the size of the corresponding stack does not exceed the available memory. This constraint is violated in most cases when considering control-flow graphs of large-scale numerical simulation programs. The tour reversal problem also arises in adjoint programs used, for example, in the context of derivative-based nonlinear optimization, sensitivity analysis, or other, often inverse, problems. The intention is to compress the tour in order not to run out of memory. As the general optimal compression problem was proven to be NP-hard [1] and big control-flow graphs results from loops in programs we do not want to use general purpose algorithms to compress the tour. We rather want to compress the tour by finding loops and replace the redundant information by proper representation of the loops.