In a published study, the problem of using Turing reduction to solve ε-NN is studied. In other words, given a query point q, a point set P, and an approximate factor ε, the purpose is to return the approximate nearest neighbor of q in P with an approximation ratio of not more than 1+ε. Moreover, a Turing reduction algorithm with O(logn) query time complexity is proposed, where the query time is the number of times that the oracle is invoked. The comparison indicates that the O(logn) query time is the lowest compared to that of all the existing algorithms. However, the disadvantage of the proposed algorithm is that there is a factor of O((d/ε)^d) in the preprocessing time complexity and space complexity. When the number of dimensions d is high, or the approximation factor ε is small, the factor would become unacceptable. Therefore, this study revises the reduction algorithm and analyzes the expected time complexity and space complexity of the algorithm when the input point set follows the Poisson point process. As a result, the expected preprocessing time complexity is reduced to O(nlogn), and the expected space complexity is reduced to O(nlogn), while the expected query time complexity remains O(logn). In this sense, the future work raised in the published study is completed.