▁family ▁is ▁indexed ▁by ▁a ▁choice ▁of ▁random ▁ ▁and ▁ ▁where ▁ ▁is ▁a ▁d ▁dimensional ▁ ▁vector ▁with ▁entries ▁chosen ▁independently ▁from ▁a ▁stable ▁distribution ▁and ▁▁ ▁is ▁a ▁real ▁number ▁chosen ▁uniformly ▁from ▁the ▁range ▁[ 0 , r ]. ▁For ▁a ▁fixed ▁ ▁the ▁hash ▁function ▁ ▁is ▁given ▁by ▁. ▁ ▁Other ▁construction ▁methods ▁for ▁hash ▁functions ▁have ▁been ▁proposed ▁to ▁better ▁fit ▁the ▁data . ▁▁ ▁In ▁particular ▁k - me ans ▁hash ▁functions ▁are ▁better ▁in ▁practice ▁than ▁projection - based ▁hash ▁functions , ▁but ▁without ▁any ▁theoretical ▁guarantee . ▁ ▁L SH ▁algorithm ▁for ▁nearest ▁neighbor ▁search ▁ ▁One ▁of ▁the ▁main ▁applications ▁of ▁L SH ▁is ▁to ▁provide ▁a ▁method ▁for ▁efficient ▁approximate ▁nearest ▁neighbor ▁search ▁algorithms . ▁ ▁Consider ▁an ▁L SH ▁family ▁. ▁ ▁The ▁algorithm ▁has ▁two ▁main ▁parameters : ▁the ▁width ▁parameter ▁ ▁and ▁the ▁number ▁of ▁hash ▁tables ▁. ▁ ▁In ▁the ▁first ▁step , ▁we ▁define ▁a ▁new ▁family ▁ ▁of ▁hash ▁functions ▁, ▁where ▁each ▁function ▁ ▁is ▁obtained ▁by ▁concaten ating ▁ ▁functions ▁ ▁from ▁, ▁i . e ., ▁. ▁ ▁In ▁other ▁words , ▁a ▁random ▁hash ▁function ▁ ▁is ▁obtained ▁by ▁concaten ating ▁ ▁randomly ▁chosen ▁hash ▁functions ▁from ▁. ▁ ▁The ▁algorithm ▁then ▁construct s ▁ ▁hash ▁tables , ▁each ▁corresponding ▁to ▁a ▁different ▁randomly ▁chosen ▁hash ▁function ▁. ▁ ▁In ▁the ▁pre processing ▁step ▁we ▁hash ▁all ▁ ▁points ▁from ▁the ▁data ▁set ▁ ▁into ▁each ▁of ▁the ▁ ▁hash ▁tables