On the exact ordered binary decision diagram size of totally symmetric functions

Heap, M.A.

doi:10.1007/bf00971647

Cited by 13 publications

(7 citation statements)

References 2 publications

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“…We derive upper bounds for the size of the SymOBDDs that only depend on the size of the maximal blocks of pairwise symmetric variables which we call symmetry sets in the following. These results generalize the upper bounds for totally symmetric Boolean functions [14], [37], nonsymmetric Boolean functions [15], and Boolean functions with one symmetry set [25].…”

Section: Introductionsupporting

confidence: 72%

“…The detailed proof can be found in [17]. Theorem 2 generalizes the results shown by [14], [15], [37].…”

Section: Theoremsupporting

confidence: 55%

“…As already mentioned, the upper bounds proven here cover the upper bounds for nonsymmetric and totally symmetric Boolean functions proven by [14], [15], [37].…”

Section: Open Problemsupporting

confidence: 52%

“…The upper bound for nonsymmetric Boolean functions has been proven to be tight in [15]. Unfortunately, the proof presented in [14] that the upper bound for totally symmetric Boolean functions is tight is faulty. We have presented a counterexample for the proof of the crucial Lemma 2 of Heap in [17, p. 64].…”

Section: Open Problemmentioning

confidence: 99%

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Least upper bounds for the size of OBDDs using symmetry properties

Heinrich-Litan

Molitor

2000

IEEE Trans. Comput.

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ÐThis paper investigates reduced ordered binary decision diagrams (OBDD) of partially symmetric Boolean functions when using variable orders where symmetric variables are adjacent. We prove upper bounds for the size of such symmetry ordered OBDDs (SymOBDD). They generalize the upper bounds for the size of OBDDs of totally symmetric Boolean functions and nonsymmetric Boolean functions proven by Heap and Mercer [14], [15] and Wegener [37]. Experimental results based on these upper bounds show that the nontrivial symmetry sets of a Boolean function should be located either right up at the beginning or right up at the end of the variable order in order to obtain best upper bounds. Index TermsÐBinary decision diagrams, variable ordering, upper worst case bounds, partial symmetric Boolean functions.

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Section: Introductionsupporting

confidence: 72%

“…The detailed proof can be found in [17]. Theorem 2 generalizes the results shown by [14], [15], [37].…”

Section: Theoremsupporting

confidence: 55%

“…As already mentioned, the upper bounds proven here cover the upper bounds for nonsymmetric and totally symmetric Boolean functions proven by [14], [15], [37].…”

Section: Open Problemsupporting

confidence: 52%

Section: Open Problemmentioning

confidence: 99%

See 2 more Smart Citations

Least upper bounds for the size of OBDDs using symmetry properties

Heinrich-Litan

Molitor

2000

IEEE Trans. Comput.

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“…Two complete papers have been devoted to the calculation of the worst case number of nodes, the first of which is Ross et al [8]. In the second, Heap [6] presented a correction to the results in [8]. Independently, Sasao [9] derived a similar expression.…”

Section: Introductionmentioning

confidence: 98%

Average and worst case number of nodes in decision diagrams of symmetric multiple-valued functions

Butler

Herscovici

Sasao

et al. 1997

IEEE Trans. Comput.

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The complexity of the optimal variable ordering problems of shared binary decision diagrams

Tani

Hamaguchi

Yajima

1993

Algorithms and Computation

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An ordered binary decision diagram (OBDD) is a directed acyclic graph that represents a Boolean function. OBDDs are also known as special cases of oblivious read-once branching programs in the field of complexity theory. Since OBDDs have many nice properties as data structures, they have been extensively studied for decades in both theoretical and practical fields, such as VLSI design, formal verification, machine learning, and combinatorial problems. Arguably, the most crucial problem in using OBDDs is that they may vary exponentially in size depending on their variable ordering (i.e., the order in which the variable are to read) when they represent the same function. Indeed, it is NP hard to find an optimal variable ordering that minimizes an OBDD for a given function. Hence, numerous studies have sought heuristics to find an optimal variable ordering. From practical as well as theoretical points of view, it is also important to seek algorithms that output optimal solutions with lower (exponential) time complexity than trivial brute-force algorithms do. Friedman and Supowit provided a clever deterministic algorithm with time/space complexity O * (3 n ), where n is the number of variables of the function, which is much better than the trivial brute-force bound O * (n!2 n ). This paper shows that a further speedup is possible with quantum computers by demonstrating the existence of a quantum algorithm that produces a minimum OBDD together with the corresponding variable ordering in O * (2.77286 n ) time and space with an exponentially small error. Moreover, this algorithm can be adapted to constructing other minimum decision diagrams such as zero-suppressed BDDs, which provide compact representations of sparse sets and are often used in the field of discrete optimization and enumeration.

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On the exact ordered binary decision diagram size of totally symmetric functions

Cited by 13 publications

References 2 publications

Least upper bounds for the size of OBDDs using symmetry properties

Least upper bounds for the size of OBDDs using symmetry properties

Average and worst case number of nodes in decision diagrams of symmetric multiple-valued functions

The complexity of the optimal variable ordering problems of shared binary decision diagrams

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