Approximate K Nearest Neighbors in High Dimensions

Given a set P of N points in a ddimensional space, along with a query point q, it is often desirable to find k points of P that are with high probability close to q. This is the Approximate k-NearestNeighbors problem. We present two algorithms for AkNN. Both require O(Nd) preprocessing time. The first algorithm has a query time cost that is O(d+logN), while the second has a query time cost that is O(d). Both algorithms create an undirected graph on the points of P by adding edges to a linked list storing P in Hilbert order. To find approximate nearest neighbors of a query point, both algorithms perform bestfirst search on this graph. The first algorithm uses standard one dimensional indexing structures to find starting points on the graph for this search, whereas the second algorithm using random starting points. Despite the quadratic preprocessing time, our algorithms have the potential to be useful in machine learning applications where the number of query points that need to be processed is large compared to the number of points in P . The linear dependence in d of the preprocessing and query time costs of our algorithms allows them to remain effective even when dealing with high-