itive ▁has hing ▁( L SH ); ▁or ▁data - dependent ▁methods , ▁such ▁as ▁Loc ality - pres erving ▁has hing ▁( LP H ). ▁ ▁De fin itions ▁An ▁L SH ▁family ▁ ▁is ▁defined ▁for ▁a ▁metric ▁space ▁, ▁a ▁threshold ▁ ▁and ▁an ▁approximation ▁factor ▁. ▁ ▁This ▁family ▁ ▁is ▁a ▁family ▁of ▁functions ▁ ▁which ▁map ▁elements ▁from ▁the ▁metric ▁space ▁to ▁a ▁bucket ▁. ▁ ▁The ▁L SH ▁family ▁satisfies ▁the ▁following ▁conditions ▁for ▁any ▁two ▁points ▁, ▁using ▁a ▁function ▁ ▁which ▁is ▁chosen ▁uniformly ▁at ▁random : ▁ ▁if ▁, ▁then ▁ ▁( i . e ., ▁ ▁and ▁ ▁col lide ) ▁with ▁probability ▁at ▁least ▁, ▁ ▁if ▁, ▁then ▁ ▁with ▁probability ▁at ▁most ▁. ▁ ▁A ▁family ▁is ▁interesting ▁when ▁. ▁ ▁Such ▁a ▁family ▁ ▁is ▁called ▁- sens itive . ▁ ▁Alternatively ▁it ▁is ▁defined ▁with ▁respect ▁to ▁a ▁universe ▁of ▁items ▁ ▁that ▁have ▁a ▁similarity ▁function ▁. ▁An ▁L SH ▁scheme ▁is ▁a ▁family ▁of ▁hash ▁functions ▁ ▁couple d ▁with ▁a ▁probability ▁distribution ▁ ▁over ▁the ▁functions ▁such ▁that ▁a ▁function ▁ ▁chosen ▁according ▁to ▁ ▁satisfies ▁the ▁property ▁that ▁ ▁for ▁any ▁. ▁ ▁Loc ality - pres erving ▁has hing ▁A ▁local ity - pres erving ▁has hing ▁is ▁a ▁hash ▁function ▁f ▁that ▁maps ▁a ▁point ▁or ▁points ▁in ▁a ▁mult id im ensional ▁coordinate ▁space ▁to ▁a ▁scalar ▁value , ▁such ▁that ▁if ▁we ▁have ▁three ▁points ▁A , ▁B ▁and