Abstract
We consider the nearest-neighbor problem over the d-cube: given a collection of points in {0, 1}d, find the one nearest to a query point (in the L1 sense). We establish a lower bound of Ω(log log d/ log log log d) on the worst-case query time. This result holds in the cell probe model with (any amount of) polynomial storage and word-size d O(1). The same lower bound holds for the approximate version of the problem, where the answer may be any point further than the nearest neighbor by a factor as large as \( 2^{\left\lfloor {(\log d)^{1 - \varepsilon } } \right\rfloor } \) , for any fixed ε > 0.
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Chakrabarti, A., Chazelle, B., Gum, B., Lvov, A. (2003). A Lower Bound on the Complexity of Approximate Nearest-Neighbor Searching on the Hamming Cube. In: Aronov, B., Basu, S., Pach, J., Sharir, M. (eds) Discrete and Computational Geometry. Algorithms and Combinatorics, vol 25. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55566-4_14
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DOI: https://doi.org/10.1007/978-3-642-55566-4_14
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