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 [math]\displaystyle{ \{0,1\}^n }[/math] 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 [math]\displaystyle{ f: \{0,1\}^n \to Q }[/math] with [math]\displaystyle{ S \subseteq Q }[/math] and [math]\displaystyle{ card(Q) = 2^n }[/math] 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 partitionning
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
References
- ↑ 1.0 1.1 1.2 1.3 Bahar, R.I.; Frohm, E.A.; Gaona, C.M.; Hachtel, G.D.; Macii, E.; Pardo, A.; Somenzi, F. (1993). "Algebraic decision diagrams and their applications" (in en-US). Proceedings of 1993 International Conference on Computer Aided Design (ICCAD). IEEE Comput. Soc. Press. pp. 188–191. doi:10.1109/iccad.1993.580054. ISBN 0-8186-4490-7. https://doi.org/10.1109/ICCAD.1993.580054.
- ↑ 2.0 2.1 Fujita, M.; McGeer, P.C.; Yang, J.C.-Y. (1997-04-01). "Multi-Terminal Binary Decision Diagrams: An Efficient Data Structure for Matrix Representation" (in en). Formal Methods in System Design 10 (2): 149–169. doi:10.1023/A:1008647823331. ISSN 1572-8102. https://doi.org/10.1023/A:1008647823331.
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