Counterexamples to the long-standing conjecture on the complexity of BDD binary operations

Authors:
Ryo Yoshinaka;Jun Kawahara;Shuhei Denzumi;Hiroki Arimura;Shin-Ichi Minato
Affiliations:
ERATO MINATO Discrete Structure Manipulation System Project, Japan Science and Technology Agency, at Hokkaido University, Sapporo 060-0814, Japan;ERATO MINATO Discrete Structure Manipulation System Project, Japan Science and Technology Agency, at Hokkaido University, Sapporo 060-0814, Japan;Graduate School of Information Science and Technology, Hokkaido University, Sapporo 060-0814, Japan;Graduate School of Information Science and Technology, Hokkaido University, Sapporo 060-0814, Japan;Graduate School of Information Science and Technology, Hokkaido University, Sapporo 060-0814, Japan and ERATO MINATO Discrete Structure Manipulation System Project, Japan Science and Technology Ag ...
Venue:
Information Processing Letters
Year:
2012

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A simpler counterexample to a long-standing conjecture on the complexity of Bryant's apply algorithm

Information Processing Letters

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Abstract

In this article, we disprove the long-standing conjecture, proposed by R.E. Bryant in 1986, that his binary decision diagram (BDD) algorithm computes any binary operation on two Boolean functions in linear time in the input-output sizes. We present Boolean functions for which the time required by Bryant@?s algorithm is a quadratic of the input-output sizes for all nontrivial binary operations, such as @?, @?, and @?. For the operations @? and @?, we show an even stronger counterexample where the output BDD size is constant, but the computation time is still a quadratic of the input BDD size. In addition, we present experimental results to support our theoretical observations.