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).^[1][2] The terminal nodes may take any value from a set of constants S.

Definition

An ADD represents a Boolean function from ${\displaystyle \{0,1\}^{n}}$ 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 ${\displaystyle f:\{0,1\}^{n}\to Q}$ with ${\displaystyle S\subseteq Q}$ and ${\displaystyle card(Q)=2^{n}}$ for some integer n. Therefore, the theorems of the Boolean algebra applies to ADD, notably the Boole's expansion theorem.^[1]

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):

• merge any isomorphic subgraphs, and
• eliminate any node whose two children are isomorphic.

ADDs are canonical according to a particular variable ordering.

Matrix partitioning

An ADD can be represented by a matrix according to its cofactors.^[2][1]

Applications

ADDs were first implemented for sparse matrix multiplication and shortest path algorithms (Bellman-Ford, Repeated Squaring, and Floyd-Warshall procedures).^[1]

See also

• Binary decision diagram
• Zero-suppressed decision diagram