Approximate Direct and Reverse Nearest Neighbor Queries, and the k-nearest Neighbor Graph

Retrieving the \emph{k-nearest neighbors} of a query object is a basic primitive in similarity searching. A related, far less explored primitive is to obtain the dataset elements which would have the query object within their own \emph{k}-nearest neighbors, known as the \emph{reverse k-nearest neighbor} query. We already have indices and algorithms to solve \emph{k}-nearest neighbors queries in general metric spaces; yet, in many cases of practical interest they degenerate to sequential scanning. The naive algorithm for reverse \emph{k}-nearest neighbor queries has quadratic complexity, because the \emph{k}-nearest neighbors of all the dataset objects must be found; this is too expensive. Hence, when solving these primitives we can tolerate trading correctness in the solution for searching time. In this paper we propose an efficient approximate approach to solve these similarity queries with high retrieval rate. Then, we show how to use our approximate \emph{k}-nearest neighbor queries to construct (an approximation of) the \emph{k-nearest neighbor graph} when we have a fixed dataset. Finally, combining both primitives we show how to \emph{dynamically maintain} the approximate \emph{k}-nearest neighbor graph of the objects currently stored within the metric dataset, that is, considering both object insertions and deletions.