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In this paper we examined an algorithm for the All-k-Nearest-Neighbor problemproposed in 1980s, which was claimed to have an $O(n\log{n})$ upper bound onthe running time. We find the algorithm actually exceeds the so claimed upperbound, and prove that it has an $\Omega(n^2)$ lower bound on the timecomplexity. Besides, we propose a new algorithm that truly achieves the$O(n\log{n})$ bound. Detailed and rigorous theoretical proofs are provided toshow the proposed algorithm runs exactly in $$O(n\log{n})$$ time.

A True $O(n \log{n}) $ Algorithm for the All-k-Nearest-Neighbors Problem