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This paper proposes a new algorithm for reducing Approximate Nearest Neighbor problem to Approximate Near Neighbor problem. The advantage of this algorithm is that it achieves (O(log {n})) query time. As a reduction problem, the query time complexity is the times of invoking the algorithm for Approximate Near Neighbor problem. All former algorithms for the same reduction need polylog(n) query time. A box split method proposed by Vaidya is used in our paper to achieve the (O(log {n})) query time complexity.

An Algorithm for Reducing Approximate Nearest Neighbor to Approximate Near Neighbor with $$O(\log {n})$$ Query Time