Finding the optimal variable ordering for binary decision diagrams

Friedman, Steven J.; Supowit, Kenneth J.

doi:10.1145/37888.37941

Cited by 119 publications

(87 citation statements)

References 6 publications

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Section: Selecting the Best Orderingmentioning

confidence: 99%

“…From machine learning, we use many of the techniques developed for the induction of decision trees (Quinlan, 1986) as well as the constructive induction algorithms first studied by Pagallo and Haussler (1990). From the logic synthesis field, we use the vast array of techniques developed for the manipulation of reduced ordered decision graphs as canonical representations for Boolean functions (Bryant, 1986;Brace et al, 1989) and the variable reordering algorithms studied by a number of different authors (Friedman and Supowit, 1990;Rudell, 1993). For the benefit of readers not familiar with the use of reduced ordered decision graphs as a tool for the manipulation of Boolean functions, Appendix A gives an overview of the techniques available and their relation to this work.…”

Section: Introductionmentioning

confidence: 99%

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Using the minimum description length principle to infer reduced ordered decision graphs

Oliveira

Sangiovanni‐Vincentelli

1996

Mach Learn

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Abstract.We propose an algorithm for the inference of decision graphs from a set of labeled instances. In particular, we propose to infer decision graphs where the variables can only be tested in accordance with a given order and no redundant nodes exist. This type of graphs, reduced ordered decision graphs, can be used as canonical representations of Boolean functions and can be manipulated using algorithms developed for that purpose. This work proposes a local optimization algorithm that generates compact decision graphs by performing local changes in an existing graph until a minimum is reached. The algorithm uses Rissanen's minimum description length principle to control the tradeoffbetween accuracy in the training set and complexity of the description. Techniques for the selection of the initial decision graph and for the selection of an appropriate ordering of the variables are also presented. Experimental results obtained using this algorithm in two sets of examples are presented and analyzed.

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Section: Selecting the Best Orderingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Using the minimum description length principle to infer reduced ordered decision graphs

Oliveira

Sangiovanni‐Vincentelli

1996

Mach Learn

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“…Regrettably, selecting the optimal ordering for a given function is NP-complete (Tani et al, 1993) and cannot be solved exactly in most cases. For this reason, and because it is a problem of high practical interest in logic synthesis, many heuristic algorithms have been proposed for this problem (Friedman & Supowit, 1990). In our setting, the problem is even more complex because we wish to select the ordering that minimizes the final RODG and this ordering may not be the same as the one that minimizes the RODG obtained after the initialization step.…”

Section: Selecting the Best Orderingmentioning

confidence: 99%

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Oliveira

Sangiovanni‐Vincentelli

1996

Machine Learning

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Abstract.We propose an algorithm for the inference of decision graphs from a set of labeled instances. In particular, we propose to infer decision graphs where the variables can only be tested in accordance with a given order and no redundant nodes exist. This type of graphs, reduced ordered decision graphs, can be used as canonical representations of Boolean functions and can be manipulated using algorithms developed for that purpose. This work proposes a local optimization algorithm that generates compact decision graphs by performing local changes in an existing graph until a minimum is reached. The algorithm uses Rissanen's minimum description length principle to control the tradeoff between accuracy in the training set and complexity of the description. Techniques for the selection of the initial decision graph and for the selection of an appropriate ordering of the variables are also presented. Experimental results obtained using this algorithm in two sets of examples are presented and analyzed.

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“…Its solution gives the optimal variable ordering and the number of minimal nodes needed. In contrast to all other exact BDD minimization techniques (see Ebendt et al 2003 for an overview) which are based on the classic method by Friedman and Supowit (1987), our approach does not need to build a BDD explicitly. With the help of this 0/1 IP formulation and the techniques for counting 0/1 vertices described in (Behle and Eisenbrand 2007) we are able to compute the variable ordering spectrum of a threshold function.…”

Section: Introductionmentioning

confidence: 99%

On threshold BDDs and the optimal variable ordering problem

Behle

2007

J Comb Optim

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Many combinatorial optimization problems can be formulated as 0/1 integer programs (0/1 IPs). The investigation of the structure of these problems raises the following tasks: count or enumerate the feasible solutions and find an optimal solution according to a given linear objective function. All these tasks can be accomplished using binary decision diagrams (BDDs), a very popular and effective datastructure in computational logics and hardware verification.We present a novel approach for these tasks which consists of an output-sensitive algorithm for building a BDD for a linear constraint (a so-called threshold BDD) and a parallel AND operation on threshold BDDs. In particular our algorithm is capable of solving knapsack problems, subset sum problems and multidimensional knapsack problems.BDDs are represented as a directed acyclic graph. The size of a BDD is the number of nodes of its graph. It heavily depends on the chosen variable ordering. Finding the optimal variable ordering is an NP-hard problem. We derive a 0/1 IP for finding an optimal variable ordering of a threshold BDD. This 0/1 IP formulation provides the basis for the computation of the variable ordering spectrum of a threshold function.We introduce our new tool azove 2.0 as an enhancement to azove 1.1 which is a tool for counting and enumerating 0/1 points. Computational results on benchmarks from the literature show the strength of our new method.Keywords Binary decision diagram · Threshold BDD · Knapsack · 0/1 integer programming · Optimal variable ordering · Variable ordering spectrum M. Behle ( )

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Finding the optimal variable ordering for binary decision diagrams

Cited by 119 publications

References 6 publications

Using the minimum description length principle to infer reduced ordered decision graphs

Using the minimum description length principle to infer reduced ordered decision graphs

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On threshold BDDs and the optimal variable ordering problem

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