Size bound of CFLOBDDs versus BDDs under different variable orderings

Determine tight upper and lower bounds on the size of a Context-Free-Language Ordered Binary Decision Diagram (CFLOBDD) for a Boolean function in terms of the size of a Binary Decision Diagram (BDD) for the same function when the CFLOBDD is permitted to use a variable ordering different from the BDD’s ordering. Ascertain whether a general bound such as O(|B|^2), O(|B|), or a sublinear function in |B| is achievable for the CFLOBDD size |C| relative to the BDD size |B|, or whether larger separations are unavoidable.

Background

The paper establishes that when a CFLOBDD and a BDD for the same Boolean function use the same variable ordering, the CFLOBDD’s size is bounded by O(|B|^3), and this bound is tight. This result is derived via structural relationships between BDDs and CFLOBDDs, including a mapping from BDD nodes to CFLOBDD groupings, and polynomial bounds on groupings, vertices, and edges.

The authors then pose the unresolved question of how the bound changes when the CFLOBDD is allowed to adopt a different variable ordering than the BDD. They explicitly ask whether stronger bounds—potentially quadratic, linear, or even sublinear in |B|—can be achieved under this flexibility, and state that the problem remains open.