Algebraic decision diagram

An algebraic decision diagram (ADD) or a multi-terminal binary decision diagram (MTBDD), is a data structure that is used to symbolically represent a Boolean function whose codomain is an arbitrary finite set S. An ADD is an extension of a reduced ordered binary decision diagram, or commonly named binary decision diagram (BDD) in the literature, which terminal nodes are not restricted to the Boolean values 0 (FALSE) and 1 (TRUE). The terminal nodes may take any value from a set of constants S.

Definition
An ADD represents a Boolean function from $$\{0,1\}^n$$ to a finite set of constants S, or carrier of the algebraic structure. An ADD is a rooted, directed, acyclic graph, which has several nodes, like a BDD. However, an ADD can have more than two terminal nodes which are elements of the set S, unlike a BDD.

An ADD can also be seen as a Boolean function, or a vectorial Boolean function, by extending the codomain of the function, such that $$f: \{0,1\}^n \to Q $$ with $$S \subseteq Q$$ and $$card(Q) = 2^n$$ for some integer n. Therefore, the theorems of the Boolean algebra applies to ADD, notably the Boole's expansion theorem.

Each node of is labeled by a Boolean variable and has two outgoing edges: a 1-edge which represents the evaluation of the variable to the value TRUE, and a 0-edge for its evaluation to FALSE.

An ADD employs the same reduction rules as a BDD (or Reduced Ordered BDD):

ADDs are canonical according to a particular variable ordering.
 * merge any isomorphic subgraphs, and
 * eliminate any node whose two children are isomorphic.

Matrix partitioning
An ADD can be represented by a matrix according to its cofactors.

Applications
ADDs were first implemented for sparse matrix multiplication and shortest path algorithms (Bellman-Ford, Repeated Squaring, and Floyd-Warshall procedures).