Randomized algorithms for comparison-based search

This paper addresses the problem of finding the nearest neighbor (or one of the R-nearest neighbors) of a query object q in a database of n objects, when we can only use a comparison oracle. The comparison oracle, given two reference objects and a query object, returns the reference object most similar to the query object. The main problem we study is how to search the database for the nearest neighbor (NN) of a query, while minimizing the questions. The difficulty of this problem depends on properties of the underlying database. We show the importance of a characterization: combinatorial disorder D which defines approximate triangle inequalities on ranks. We present a lower bound of Ω(Dlog n/D + D²) average number of questions in the search phase for any randomized algorithm, which demonstrates the fundamental role of D for worst case behavior. We develop a randomized scheme for NN retrieval in O(D³ log² n + Dlog² n log log n^D3 ) questions. The learning requires asking O(nD³ log² n + Dlog² n log log n^D3 ) questions and O(n log² n/ log(2D)) bits to store.