Quest for Fast Partial Search Algorithm
  Quantum algorithm can find target item in a database faster than any classical. One can trade accuracy for speed and find a part of the database (a block) containing the target item even faster: this is partial search. One can think of partial search in following terms: an exact address of the target item is given by a sequence of many bites, but we want to find only several first bites of the address. We consider different partial search algorithms consisting of a sequence of global and local searches and suggest the optimal one. Global search consists of the standard Grover iterations for the whole database. Local search consists of Grover iterations in each individual block made simultaneously in all blocks. Efficiency of an algorithm is measured by number of queries to the oracle. 
Vladimir Korepin and Jinfeng Liao