| We present a randomized algorithm for the approximate nearest neighbor problem in d-dimensional Euclidean space. Given N points {xj} in Rd , the algorithm attempts to find k nearest neighbors for each of xj, where k is a user-specified integer parameter. The algorithm is iterative, and its CPU time requirements are proportional to T · N · (d · (log d) + k · (d + log k) · (log N)) + N · k 2 · (d + log k), with T the number of iterations performed. The memory requirements of the procedure are of the order N · (d + k).;A byproduct of the scheme is a data structure, permitting a rapid search for the k nearest neighbors among {xj} for an arbitrary point x ∈ Rd . The cost of each such query is proportional to T · (d · (log d) + log(N/k) · k · (d + log k)), and the memory requirements for the requisite data structure are of the order N · (d + k) + T · (d + N).;The algorithm utilizes random rotations and a basic divide-and-conquer scheme, followed by a local graph search. We analyze the scheme's behavior for certain types of distributions of {xj}, and illustrate its performance via several numerical examples. |
