, ▁ 0 1 , ▁and ▁ 0 0 ▁to ▁denote ▁symbols ▁ 1 , ▁ 0 , ▁and ▁– ▁respectively , ▁one ▁can ▁represent ▁the ▁above ▁set ▁with ▁bit ▁vectors ▁in ▁the ▁form ▁of ▁. ▁Notice ▁that ▁the ▁set ▁of ▁bit ▁vectors ▁is ▁sparse , ▁in ▁that ▁the ▁number ▁of ▁vectors ▁is ▁fewer ▁than ▁ 2 , ▁which ▁is ▁the ▁maximum ▁number ▁of ▁bit ▁vectors , ▁and ▁the ▁set ▁contains ▁many ▁elements ▁equal ▁to ▁zero . ▁In ▁this ▁case , ▁a ▁node ▁can ▁be ▁omitted ▁if ▁setting ▁the ▁node ▁variable ▁to ▁ 1 ▁causes ▁the ▁function ▁to ▁yield ▁ 0 . ▁This ▁is ▁seen ▁in ▁the ▁condition ▁that ▁a ▁ 1 ▁at ▁some ▁bit ▁position ▁implies ▁that ▁the ▁vector ▁is ▁not ▁in ▁the ▁set . ▁For ▁sparse ▁sets , ▁this ▁condition ▁is ▁common , ▁and ▁hence ▁many ▁node ▁elimin ations ▁are ▁possible . ▁▁ ▁Min ato ▁has ▁proved ▁that ▁Z DD s ▁are ▁especially ▁suitable ▁for ▁combin atorial ▁problems , ▁such ▁as ▁the ▁classical ▁problems ▁in ▁two - level ▁logic ▁minim ization , ▁kn ight ’ s ▁tour ▁problem , ▁fault ▁simulation , ▁timing ▁analysis , ▁the ▁N - que ens ▁problem , ▁as ▁well ▁as ▁weak ▁division . ▁By ▁using ▁Z DD s , ▁one ▁can ▁reduce ▁the ▁size ▁of ▁the ▁representation ▁of ▁a ▁set ▁of ▁n - bit ▁vectors ▁in ▁O B DD s ▁by ▁at ▁most ▁a ▁factor ▁of ▁n . ▁In ▁practice , ▁the ▁optimization ▁is ▁statist ically ▁significant . ▁ ▁De fin itions ▁We ▁define ▁a ▁Zero