A True O(n log n) algorithm for the all-k-nearest-neighbors problem

Abstract

In this paper we examined an algorithm for the All-k-Nearest-Neighbor problem proposed in 1980s, which was claimed to have an upper bound on the running time. We find the algorithm actually exceeds the so claimed upper bound, and prove that it has an lower bound on the time complexity. Besides, we propose a new algorithm that truly achieves the bound. Detailed and rigorous theoretical proofs are provided to show the proposed algorithm runs exactly in time.