Theoretical work on high-dimensional nearest neighbor search has focused on the setting where a single point is sought within a known search radius, and an acceptable approximation ratio c is given. Locality Sensitive Hashing is a powerful framework for addressing this problem. In practice one usually seeks the (exact) k nearest points, the search radius is unknown, and the parameter c must be chosen in a way that depends on the data distribution. Though reductions of the latter problem to the former exist, they incur polylogarithmic overhead in time and/or space, which in turn make them unattractive in many practical settings. We address this discrepancy between theory and practice by suggesting new, simple, more efficient reductions for solving the k-Nearest Neighbor search problem using Locality Sensitive Hashing. Joint work with Tobias Christiani and Mikkel Thorup.