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39
Fast Exact Minimization of BDDs
 In Design Automation Conf
, 1998
"... We present a new exact algorithm for #nding the optimal variable ordering for reduced ordered Binary Decision Diagrams #BDDs#. The algorithm makes use of a lower bound technique known from VLSI design. Up to now this technique has been used only for theoretical considerations and it is adapted here ..."
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Cited by 26 (9 self)
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We present a new exact algorithm for #nding the optimal variable ordering for reduced ordered Binary Decision Diagrams #BDDs#. The algorithm makes use of a lower bound technique known from VLSI design. Up to now this technique has been used only for theoretical considerations and it is adapted here for our purpose. Furthermore, the algorithm supports symmetry aspects and makes use of a hashing based data structure. Experimental results are given to demonstrate the e#ciency of our approach. We succeeded in minimizing adder functions with up to 64 variables, while all other previously presented approaches fail. 1 Introduction Recently, several design methods have been proposed that are based on ordered Binary Decision Diagrams #BDDs# #7#. The resulting circuits have very nice properties, like e.g. testability #2, 1# and lowpower #17#. For synthesis approaches based on Pass Transistor Logic #PTL# BDDs seem to be a good starting point. First promising results on how to transform a decisi...
On the Existence of Polynomial Time Approximation Schemes for OBDD Minimization
 STACS'98, LNCS 1373
, 1998
"... Abstract 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 ¡£¢complete. We strengthen this result ..."
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Cited by 19 (3 self)
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Abstract 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 ¡£¢complete. We strengthen this result by showing that there in no polynomial time approximation scheme for the variable ordering problem unless ¢¥¤¥¡£¢. We also prove a small lower bound on the performance ratio of a polynomial time approximation algorithm under the assumption ¢§ ¦ ¤¥¡£¢
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 19 (6 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...
Local Encoding Transformations for Optimizing OBDDRepresentations of Finite State Machines
 In FMCAD
, 1996
"... Ordered binary decision diagrams are the stateoftheart representation of switching functions. In order to keep the sizes of the OBDDs tractable, heuristics and dynamic reordering algorithms are applied to optimize the underlying variable order. When finite state machines are represented by OBDDs ..."
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Cited by 18 (8 self)
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Ordered binary decision diagrams are the stateoftheart representation of switching functions. In order to keep the sizes of the OBDDs tractable, heuristics and dynamic reordering algorithms are applied to optimize the underlying variable order. When finite state machines are represented by OBDDs the state encoding can be used as an additional optimization parameter. In this paper, we analyze local encoding transformations which can be applied dynamically. First, we investigate the potential of reencoding techniques. We then propose the use of an XORtransformation and show why this transformation is most suitable among the set of all encoding transformations. Preliminary experimental results illustrate that the proposed method in fact yields a reduction of the OBDDsizes. 1 Introduction Ordered binary decision diagrams (OBDDs) which have been introduced by Bryant [Bry86] provide an efficient graphbased data structure for switching functions. The main optimization parameter of OBD...
Learning Heuristics for OBDD Minimization by Evolutionary Algorithms
 IN PARALLEL PROBLEM SOLVING FROM NATURE, LNCS 1141
, 1996
"... Ordered Binary Decision Diagrams (OBDDs) are the stateoftheart data structure in CAD for ICs. OBDDs are very sensitive to the chosen variable ordering, i.e. the size may vary from linear to exponential. In this paper we present an Evolutionary Algorithm (EA) that learns good heuristics for OB ..."
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Cited by 15 (6 self)
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Ordered Binary Decision Diagrams (OBDDs) are the stateoftheart data structure in CAD for ICs. OBDDs are very sensitive to the chosen variable ordering, i.e. the size may vary from linear to exponential. In this paper we present an Evolutionary Algorithm (EA) that learns good heuristics for OBDD minimization starting from a given set of basic operations. The difference to other previous approaches to OBDD minimization is that the EA does not solve the problem directly. Rather, it developes strategies for solving the problem. To demonstrate the efficiency of our approach experimental results are given. The newly developed heuristics are more efficient than other previously presented methods.
CONTEXTSENSITIVE POINTER ANALYSIS USING BINARY DECISION DIAGRAMS
, 2007
"... in my opinion, it ..."
Learning to Order BDD Variables in Verification
 Journal of Artificial Intelligence Research
, 2003
"... The size and complexity of software and hardware systems have significantly increased in the past years. As a result, it is harder to guarantee their correct behavior. One of the most successful methods for automated verification of finitestate systems is model checking. Most of the current mode ..."
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Cited by 12 (0 self)
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The size and complexity of software and hardware systems have significantly increased in the past years. As a result, it is harder to guarantee their correct behavior. One of the most successful methods for automated verification of finitestate systems is model checking. Most of the current modelchecking systems use binary decision diagrams (BDDs) for the representation of the tested model and in the verification process of its properties.
Complexity Theoretical Aspects of OFDDs
, 1996
"... We extend the list of complexity results for OFDDs on problems arising in practical applications. We show that it is NPhard to decide whether a function represented by some OFDD can be represented by an OFDD of size s using another variable ordering. Given an OFDD representation for an incompletely ..."
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Cited by 9 (2 self)
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We extend the list of complexity results for OFDDs on problems arising in practical applications. We show that it is NPhard to decide whether a function represented by some OFDD can be represented by an OFDD of size s using another variable ordering. Given an OFDD representation for an incompletely defined function, it is NPhard to compute an optimal OFDD cover for this function respecting the same variable ordering. The replacement of variables by constants may cause an exponential blowup of the OFDD size. Finally, it is investigated how a local change of the variable ordering may change the OFDD size. This leads to simulated annealing algorithms to improve variable orderings. I. Introduction OBDDs (ordered binary decision diagrams) introduced by Bryant [5] are the most common representation of Boolean functions in many applications, in particular in hardware verification and model checking. Most of the complexity theoretic problems concerning OBDDs are already solved. Kebschull, ...