1 , 4 } ▁are ▁not ▁frequent , ▁any ▁larger ▁set ▁which ▁contains ▁{ 1 , 3 } ▁or ▁{ 1 , 4 } ▁cannot ▁be ▁frequent . ▁In ▁this ▁way , ▁we ▁can ▁pr une ▁sets : ▁we ▁will ▁now ▁look ▁for ▁frequent ▁tri ples ▁in ▁the ▁database , ▁but ▁we ▁can ▁already ▁exclude ▁all ▁the ▁tri ples ▁that ▁contain ▁one ▁of ▁these ▁two ▁pairs : ▁ ▁in ▁the ▁example , ▁there ▁are ▁no ▁frequent ▁triple ts . ▁{ 2 , 3 , 4 } ▁is ▁below ▁the ▁minimal ▁threshold , ▁and ▁the ▁other ▁triple ts ▁were ▁ex cluded ▁because ▁they ▁were ▁super ▁sets ▁of ▁pairs ▁that ▁were ▁already ▁below ▁the ▁threshold . ▁ ▁We ▁have ▁thus ▁determined ▁the ▁frequent ▁sets ▁of ▁items ▁in ▁the ▁database , ▁and ▁illustrated ▁how ▁some ▁items ▁were ▁not ▁counted ▁because ▁one ▁of ▁their ▁subsets ▁was ▁already ▁known ▁to ▁be ▁below ▁the ▁threshold . ▁ ▁Lim itations ▁▁ ▁Ap ri ori , ▁while ▁histor ically ▁significant , ▁suff ers ▁from ▁a ▁number ▁of ▁in e ffic ien cies ▁or ▁trade - offs , ▁which ▁have ▁spawn ed ▁other ▁algorithms . ▁ ▁C andid ate ▁generation ▁generates ▁large ▁numbers ▁of ▁subsets ▁( the ▁algorithm ▁attempts ▁to ▁load ▁up ▁the ▁candidate ▁set ▁with ▁as ▁many ▁as ▁possible ▁before ▁each ▁scan ). ▁ ▁B ottom - up ▁subset ▁expl oration ▁( ess entially ▁a ▁bread th - first ▁travers al ▁of ▁the ▁subset ▁lattice ) ▁ ▁finds ▁any ▁maximal ▁subset ▁S ▁only ▁after ▁all ▁ ▁of ▁its ▁proper ▁subsets .