Quantum Algorithm for Finding the Optimal Variable Ordering for Binary Decision Diagrams

Author Seiichiro Tani

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Seiichiro Tani
  • NTT Communication Science Laboratories, NTT Corporation, Atsugi, Japan

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Seiichiro Tani. Quantum Algorithm for Finding the Optimal Variable Ordering for Binary Decision Diagrams. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 36:1-36:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


An ordered binary decision diagram (OBDD) is a directed acyclic graph that represents a Boolean function. 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 (Very Large Scale Integration) 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 variables are to be 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. Friedman and Supowit provided a clever deterministic algorithm with time/space complexity O^∗(3ⁿ), where n is the number of variables of the function, which is much better than the trivial brute-force bound O^∗(n!2ⁿ). This paper shows that a further speedup is possible with quantum computers by presenting a quantum algorithm that produces a minimum OBDD together with the corresponding variable ordering in O^∗(2.77286ⁿ) time and space with an exponentially small error probability. Moreover, this algorithm can be adapted to constructing other minimum decision diagrams such as zero-suppressed BDDs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Theory of computation → Data structures design and analysis
  • Binary Decision Diagram
  • Variable Ordering
  • Quantum Algorithm


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