▁defines ▁a ▁single ▁hash ▁function ▁ ▁mapping ▁input ▁sets ▁to ▁elements ▁of ▁. ▁ ▁Define ▁the ▁function ▁family ▁ ▁to ▁be ▁the ▁set ▁of ▁all ▁such ▁functions ▁and ▁let ▁ ▁be ▁the ▁uniform ▁distribution . ▁Given ▁two ▁sets ▁ ▁the ▁event ▁that ▁ ▁corresponds ▁exactly ▁to ▁the ▁event ▁that ▁the ▁minim izer ▁of ▁ ▁over ▁ ▁lies ▁inside ▁. ▁As ▁ ▁was ▁chosen ▁uniformly ▁at ▁random , ▁ ▁and ▁ ▁define ▁an ▁L SH ▁scheme ▁for ▁the ▁J acc ard ▁index . ▁ ▁Because ▁the ▁symmetric ▁group ▁on ▁ ▁elements ▁has ▁size ▁! , ▁choosing ▁a ▁truly ▁random ▁permut ation ▁from ▁the ▁full ▁symmetric ▁group ▁is ▁in fe as ible ▁for ▁even ▁moder ately ▁s ized ▁. ▁Because ▁of ▁this ▁fact , ▁there ▁has ▁been ▁significant ▁work ▁on ▁finding ▁a ▁family ▁of ▁permut ations ▁that ▁is ▁" min - wise ▁independent " ▁— ▁a ▁permut ation ▁family ▁for ▁which ▁each ▁element ▁of ▁the ▁domain ▁has ▁equal ▁probability ▁of ▁being ▁the ▁minimum ▁under ▁a ▁randomly ▁chosen ▁. ▁It ▁has ▁been ▁established ▁that ▁a ▁min - wise ▁independent ▁family ▁of ▁permut ations ▁is ▁at ▁least ▁of ▁size ▁, ▁and ▁that ▁this ▁bound ▁is ▁tight . ▁ ▁Because ▁min - wise ▁independent ▁families ▁are ▁too ▁big ▁for ▁practical ▁applications , ▁two ▁variant ▁not ions ▁of ▁min - wise ▁independence ▁are ▁introduced : ▁restricted ▁min - wise ▁independent ▁permut ations ▁families , ▁and ▁approximate ▁min - wise ▁independent ▁families . ▁Rest rict ed ▁min - wise ▁independence ▁is ▁the ▁min - wise ▁independence ▁property ▁restricted ▁to ▁certain