 Binary decision diagram

In the field of computer science, a binary decision diagram (BDD) or branching program, like a negation normal form (NNF) or a propositional directed acyclic graph (PDAG), is a data structure that is used to represent a Boolean function. On a more abstract level, BDDs can be considered as a compressed representation of sets or relations. Unlike other compressed representations, operations are performed directly on the compressed representation, i.e. without decompression.
Contents
Definition
A Boolean function can be represented as a rooted, directed, acyclic graph, which consists of decision nodes and two terminal nodes called 0terminal and 1terminal. Each decision node is labeled by a Boolean variable and has two child nodes called low child and high child. The edge from a node to a low (high) child represents an assignment of the variable to 0 (1). Such a BDD is called 'ordered' if different variables appear in the same order on all paths from the root. A BDD is said to be 'reduced' if the following two rules have been applied to its graph:
 Merge any isomorphic subgraphs.
 Eliminate any node whose two children are isomorphic.
In popular usage, the term BDD almost always refers to Reduced Ordered Binary Decision Diagram (ROBDD in the literature, used when the ordering and reduction aspects need to be emphasized).^{[1]} One advantage of an ROBDD is that it is canonical (unique) for a particular function and variable order.^{[2]} This property makes it useful in functional equivalence checking and other operations like functional technology mapping.
A path from the root node to the 1terminal represents a (possibly partial) variable assignment for which the represented Boolean function is true. As the path descends to a low child (high child) from a node, then that node's variable is assigned to 0 (1).
Example
The left figure below shows a binary decision tree (the reduction rules are not applied), and a truth table, each representing the function f (x1, x2, x3). In the tree on the left, the value of the function can be determined for a given variable assignment by following a path down the graph to a terminal. In the figures below, dotted lines represent edges to a low child, while solid lines represent edges to a high child. Therefore, to find (x1=0, x2=1, x3=1), begin at x1, traverse down the dotted line to x2 (since x1 has an assignment to 0), then down two solid lines (since x2 and x3 each have an assignment to one). This leads to the terminal 1, which is the value of f (x1=0, x2=1, x3=1).
The binary decision tree of the left figure can be transformed into a binary decision diagram by maximally reducing it according to the two reduction rules. The resulting BDD is shown in the right figure.
History
The basic idea from which the data structure was created is the Shannon expansion. A switching function is split into two subfunctions (cofactors) by assigning one variable (cf. ifthenelse normal form). If such a subfunction is considered as a subtree, it can be represented by a binary decision tree. Binary decision diagrams (BDD) were introduced by Lee,^{[3]} and further studied and made known by Akers^{[4]} and Boute.^{[5]}
The full potential for efficient algorithms based on the data structure was investigated by Randal Bryant at Carnegie Mellon University: his key extensions were to use a fixed variable ordering (for canonical representation) and shared subgraphs (for compression). Applying these two concepts results in an efficient data structure and algorithms for the representation of sets and relations.^{[6]}^{[7]} By extending the sharing to several BDDs, i.e. one subgraph is used by several BDDs, the data structure Shared Reduced Ordered Binary Decision Diagram is defined.^{[8]} The notion of a BDD is now generally used to refer to that particular data structure.
In his video lecture Fun With Binary Decision Diagrams (BDDs), Donald Knuth calls BDDs "one of the only really fundamental data structures that came out in the last twentyfive years" and mentions that Bryant's 1986 paper was for some time one of the mostcited papers in computer science.
Applications
BDDs are extensively used in CAD software to synthesize circuits (logic synthesis) and in formal verification. There are several lesser known applications of BDD, including Fault tree analysis, Bayesian Reasoning and Product Configuration^{[citation needed]}.
Every arbitrary BDD (even if it is not reduced or ordered) can be directly implemented by replacing each node with a 2 to 1 multiplexer; each multiplexer can be directly implemented by a 4LUT in a FPGA. It is not so simple to convert from an arbitrary network of logic gates to a BDD^{[citation needed]} (unlike the andinverter graph).
Variable ordering
The size of the BDD is determined both by the function being represented and the chosen ordering of the variables. For a boolean function then depending upon the ordering of the variables we would end up getting a graph whose number of nodes would be linear (in n) at the best and exponential at the worst case. Let us consider the Boolean function Using the variable ordering , the BDD needs 2^{n+1} nodes to represent the function. Using the ordering , the BDD consists of 2n nodes.
It is of crucial importance to care about variable ordering when applying this data structure in practice. The problem of finding the best variable ordering is NPhard.^{[9]} For any constant c > 1 it is even NPhard to compute a variable ordering resulting in an OBDD with a size that is at most c times larger than an optimal one.^{[10]} However there exist efficient heuristics to tackle the problem.^{[citation needed]}
There are functions for which the graph size is always exponential — independent of variable ordering. This holds e. g. for the multiplication function (an indication^{[citation needed]} as to the apparent complexity of factorization ).
Researchers have of late suggested refinements on the BDD data structure giving way to a number of related graphs, such as BMD (Binary Moment Diagrams), ZDD (Zero Suppressed Decision Diagram), FDD (Free Binary Decision Diagrams), PDD (Parity decision Diagrams), and MTBDDs (Multiple terminal BDDs).
Logical operations on BDDs
Many logical operations on BDDs can be implemented by polynomialtime graph manipulation algorithms.
 conjunction
 disjunction
 negation
 existential abstraction
 universal abstraction
However, repeating these operations several times, for example forming the conjunction or disjunction of a set of BDDs, may in the worst case result in an exponentially big BDD. This is because any of the preceding operations for two BDDs may result in a BDD with a size proportional to the product of the BDDs' sizes, and consequently for several BDDs the size may be exponential.
See also
 Boolean satisfiability problem
 L/poly, a complexity class that captures the complexity of problems with polynomially sized BDDs
 Model checking
 Radix tree
 Binary key – a method of species identification in biology using binary trees
 Barrington's theorem
References
 ^ The Art of Computer Programming, vol 4A, Donald E. Knuth
 ^ GraphBased Algorithms for Boolean Function Manipulation, Randal E. Bryant, 1986
 ^ C. Y. Lee. "Representation of Switching Circuits by BinaryDecision Programs". Bell Systems Technical Journal, 38:985–999, 1959.
 ^ Sheldon B. Akers. Binary Decision Diagrams, IEEE Transactions on Computers, C27(6):509–516, June 1978.
 ^ Raymond T. Boute, "The Binary Decision Machine as a programmable controller". EUROMICRO Newsletter, Vol. 1(2):16–22, January 1976.
 ^ Randal E. Bryant. "GraphBased Algorithms for Boolean Function Manipulation". IEEE Transactions on Computers, C35(8):677–691, 1986.
 ^ R. E. Bryant, "Symbolic Boolean Manipulation with Ordered Binary Decision Diagrams", ACM Computing Surveys, Vol. 24, No. 3 (September, 1992), pp. 293–318.
 ^ Karl S. Brace, Richard L. Rudell and Randal E. Bryant. "Efficient Implementation of a BDD Package". In Proceedings of the 27th ACM/IEEE Design Automation Conference (DAC 1990), pages 40–45. IEEE Computer Society Press, 1990.
 ^ Beate Bollig, Ingo Wegener. Improving the Variable Ordering of OBDDs Is NPComplete, IEEE Transactions on Computers, 45(9):993–1002, September 1996.
 ^ Detlef Sieling. "The nonapproximability of OBDD minimization." Information and Computation 172, 103–138. 2002.
 R. Ubar, "Test Generation for Digital Circuits Using Alternative Graphs (in Russian)", in Proc. Tallinn Technical University, 1976, No.409, Tallinn Technical University, Tallinn, Estonia, pp. 75–81.
Further reading
 D. E. Knuth, "The Art of Computer Programming Volume 4, Fascicle 1: Bitwise tricks & techniques; Binary Decision Diagrams" (Addison–Wesley Professional, March 27, 2009) viii+260pp, ISBN 0321580508. Draft of Fascicle 1b available for download.
 H. R. Andersen "An Introduction to Binary Decision Diagrams," Lecture Notes, 1999, IT University of Copenhagen.
 Ch. Meinel, T. Theobald, "Algorithms and Data Structures in VLSIDesign: OBDD – Foundations and Applications", SpringerVerlag, Berlin, Heidelberg, New York, 1998. Complete textbook available for download.
 Rüdiger Ebendt; Görschwin Fey; Rolf Drechsler (2005). Advanced BDD optimization. Springer. ISBN 9780387254531.
 Bernd Becker; Rolf Drechsler (1998). Binary Decision Diagrams: Theory and Implementation. Springer. ISBN 9781441950475.
External links
Available OBDD Packages
 ABCD: The ABCD package by Armin Biere, Johannes Kepler Universität, Linz.
 BuDDy: A BDD package by Jørn LindNielsen
 CMU BDD, BDD package, Carnegie Mellon University, Pittsburgh
 CrocoPat, BDD package and a highlevel querying language, Ecole Polytechnique Fédérale de Lausanne (EPFL), Switzerland
 CUDD: BDD package, University of Colorado, Boulder
 Biddy: multiplatform academic BDD package, University of Maribor, Slovenia
 JavaBDD, a Java port of BuDDy that also interfaces to CUDD, CAL, and JDD
 The Berkeley CAL package which does breadthfirst manipulation
 TUD BDD: A BDD package and a worldlevel package by Stefan Höreth
 Vahidi's JDD, a java library that supports common BDD and ZBDD operations
 Vahidi's JBDD, a Java interface to BuDDy and CUDD packages
 A. Costa BFunc, includes a BDD boolean logic simplifier supporting up to 32 inputs / 32 outputs (independently)
 DDD: A C++ library with support for integer valued and hierarchical decision diagrams.
 JINC: A C++ library developed at University of Bonn, Germany, supporting several BDD variants and multithreading.
 OBDD: A Haskell package for BDD
 Configit Product Modeler: A BDDbased tool for product configuration developed by Configit, Copenhagen.
Categories: Diagrams
 Graph data structures
 Model checking
 Boolean algebra
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