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The Nonapproximability of OBDD Minimization
 Information and Computation
, 1998
"... The size of Ordered Binary Decision Diagrams (OBDDs) is determined by the chosen variable ordering. A poor choice may cause an OBDD to be too large to fit into the available memory. The decision variant of the variable ordering problem is known to be NP complete. We strengthen this result by showin ..."
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Cited by 22 (5 self)
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The size of Ordered Binary Decision Diagrams (OBDDs) is determined by the chosen variable ordering. A poor choice may cause an OBDD to be too large to fit into the available memory. The decision variant of the variable ordering problem is known to be NP complete. We strengthen this result by showing that for each constant c ? 1 there is no polynomial time approximation algorithm with the performance ratio c for the variable ordering problem unless P = NP . This result justifies, also from a theoretical point of view, to use heuristics for the variable ordering problem. 1 Introduction Ordered Binary Decision Diagrams (OBDDs) are the stateoftheart data structure for Boolean functions in programs for problems like logic synthesis, model checking or circuit verification. The reason is that many functions occurring in such applications can be represented by OBDDs of reasonable size and that for operations on Boolean functions like equivalence test or synthesis with binary operators eff...
Bounds on the OBDDSize of Integer Multiplication via Universal Hashing
, 2005
"... Bryant [5] has shown that any OBDD for the function MULn−1,n, i.e. the middle bit of the nbit multiplication, requires at least 2 n/8 nodes. In this paper a stronger lower bound of essentially 2 n/2 /61 is proven by a new technique, using a universal family of hash functions. As a consequence, one ..."
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Cited by 13 (1 self)
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Bryant [5] has shown that any OBDD for the function MULn−1,n, i.e. the middle bit of the nbit multiplication, requires at least 2 n/8 nodes. In this paper a stronger lower bound of essentially 2 n/2 /61 is proven by a new technique, using a universal family of hash functions. As a consequence, one cannot hope anymore to verify e.g. 128bit multiplication circuits using OBDDtechniques because the representation of the middle bit of such a multiplier requires more than 3 · 10 17 OBDDnodes. Further, a first nontrivial upper bound of 7/3 · 2 4n/3 for the OBDDsize of MULn−1,n is provided.
Reduction of sizes of decision diagrams by autocorrelation functions
 IEEE Trans. on Computers
, 2003
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Complexity Theoretical Results for Randomized Branching Programs
, 1998
"... This work is settled in the area of complexity theory for restricted variants of branching programs. Today, branching programs can be considered one of the standard nonuniform models of computation. One reason for their popularity is that they allow to describe computations in an intuitively straigh ..."
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Cited by 8 (7 self)
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This work is settled in the area of complexity theory for restricted variants of branching programs. Today, branching programs can be considered one of the standard nonuniform models of computation. One reason for their popularity is that they allow to describe computations in an intuitively straightforward way and promise to be easier to analyze than the traditional models. In complexity theory, we are mainly interested in upper and lower bounds on the size of branching programs. Although proving superpolynomial lower bounds on the size of general branching programs still remains a challenging open problem, there has been considerable success in the study of lower bound techniques for various restricted variants, most notably perhaps readonce branching programs and OBDDs (ordered binary decision diagrams). Surprisingly, OBDDs have also turned out to be extremely useful in practical applications as a data structure for Boolean functions. So far, research has concentrated on determinis...
On Representation and Genetic Operators in Evolutionary Algorithms
 IEEE TRANS. ON EVOLUTION COMPUTATION
, 1998
"... The application of evolutionary algorithms (EAs) requires as a basic design decision the choice of a suitable representation of the variable space and appropriate genetic operators. In practice mainly problemspecific representations with specific genetic operators and miscellaneous extensions can ..."
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Cited by 7 (1 self)
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The application of evolutionary algorithms (EAs) requires as a basic design decision the choice of a suitable representation of the variable space and appropriate genetic operators. In practice mainly problemspecific representations with specific genetic operators and miscellaneous extensions can be observed. In this connection it attracts attention that hardly any formal requirements on the genetic operators are stated. In this article we first formalize the representation problem and then propose a package of requirements to guide the design of genetic operators. By the definition of distance measures on the geno and phenotype space it is possible to integrate problemspecific knowledge into the genetic operators. As an example we show how this package of requirements can be used to design a genetic programming (GP) system for finding Boolean functions.
Approximations by OBDDs and the Variable Ordering Problem
, 1999
"... . Ordered binary decision diagrams (OBDDs) and their variants are motivated by the need to represent Boolean functions in applications. Research concerning these applications leads also to problems and results interesting from theoretical point of view. In this paper, methods from communication comp ..."
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Cited by 5 (4 self)
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. Ordered binary decision diagrams (OBDDs) and their variants are motivated by the need to represent Boolean functions in applications. Research concerning these applications leads also to problems and results interesting from theoretical point of view. In this paper, methods from communication complexity and information theory are combined to prove that the direct storage access function and the inner product function have the following property. They have linear OBDD size for some variable ordering and, for most variable orderings 0 , all functions which approximate them on considerably more than half of the inputs, need exponential 0 OBDD size. These results have implications for the use of OBDDs in experiments with genetic programming. 1 INTRODUCTION Branching programs (BPs) or binary decision diagrams (BDDs), which is just another name, are representations of Boolean functions f 2 Bn , i.e., f : f0; 1g n ! f0; 1g. They are compact but not useful for manipulations o...
BDDs  Design, Analysis, Complexity, and Applications
 Discrete Applied Mathematics 138
, 2004
"... BDDs (binary decision diagrams) and their variants are the most frequently used representation types or data structures for boolean functions. Research on BDD variants has turned out to be one of the areas where the symbiosis between theoretical investigations in algorithm design and analysis, compl ..."
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Cited by 4 (0 self)
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BDDs (binary decision diagrams) and their variants are the most frequently used representation types or data structures for boolean functions. Research on BDD variants has turned out to be one of the areas where the symbiosis between theoretical investigations in algorithm design and analysis, complexity theory, and applications has led to progress in theory and in applications. Here the different roots of the interest in BDDs are described, the main BDD variants and their algorithmic properties are presented, the representation size of selected functions is investigated, lower bound techniques are discussed and applications to algorithmic graph problems and hardware verification problems are presented.
Asymptotically Optimal Bounds For OBDDs And The Solution Of Some Basic OBDD Problems
, 2000
"... Ordered binary decision diagrams (OBDDs) are nowadays the most common dynamic data structure or representation type for Boolean functions. Among the many areas of application are verification, model checking, and computer aided design. For many functions it is easy to estimate the OBDD size but asym ..."
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Cited by 3 (1 self)
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Ordered binary decision diagrams (OBDDs) are nowadays the most common dynamic data structure or representation type for Boolean functions. Among the many areas of application are verification, model checking, and computer aided design. For many functions it is easy to estimate the OBDD size but asymptotically optimal bounds are only known in simple situations. In this paper, methods for proving asymptotically optimal bounds are presented and applied to the solution of some basic problems concerning OBDDs. The largest size increase by a synthesis step of piOBDDs followed by an optimal reordering is determined as well as the largest ratio of the size of deterministic finite automata, quasireduced OBDDs, and zerosuppressed BDDs compared to the size of OBDDs. Moreover, the worst case OBDD size of functions with a given number of 1inputs is investigated.
Optimal Ordered Binary Decision Diagrams for ReadOnce Formulas
, 1999
"... In many applications like verification or combinatorial optimization, OBDDs (ordered binary decision diagrams) are used as a representation or data structure for Boolean functions. Efficient algorithms exist for the important operations on OBDDs, and many functions can be represented in reasonable s ..."
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Cited by 2 (0 self)
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In many applications like verification or combinatorial optimization, OBDDs (ordered binary decision diagrams) are used as a representation or data structure for Boolean functions. Efficient algorithms exist for the important operations on OBDDs, and many functions can be represented in reasonable size if a good variable ordering is chosen. In general, it is NPhard to compute optimal or nearoptimal variable orderings, and already simple classes of Boolean functions contain functions whose OBDD size is exponential for each variable ordering. For the class of Boolean functions representable by fanin 2 readonce formulas the structure of optimal variable orderings is described, leading to a linear time algorithm for the construction of optimal variable orderings and the size of the corresponding OBDD. Moreover, it is proved that the hardest readonce formula has an OBDD size of order n # where # = log 4 (3 + # 5) < 1.1943. Key words: Ordered binary decision diagram, efficient algorit...
Evolving binary decision diagrams with emergent variable orderings
 Parallel Problem Solving from Nature  PPSN IX, volume 4193 of LNCS
, 2006
"... Abstract. Binary Decision Diagrams (BDDs) have become the data structure of choice for representing discrete functions in some design and verification applications: They are compact and efficient to manipulate with strong theoretical underpinnings. However, and despite many appealing characteristics ..."
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Abstract. Binary Decision Diagrams (BDDs) have become the data structure of choice for representing discrete functions in some design and verification applications: They are compact and efficient to manipulate with strong theoretical underpinnings. However, and despite many appealing characteristics, BDDs are not a representation commonly considered for evolutionary computation (EC). The inherent difficulties associated with evolving graphs combined with the variable ordering problem poses a significant challenge which is yet to be overcome. This work addresses this challenge and presents a new approach to evolving BDDs that exhibits good variable orderings as an emergent property. 1