We prove new lower bounds for nearest neighbor search in the Hamming cube. Our lower bounds are for randomized, two-sided error, algorithms in Yao's cell probe model. Our bounds are in the form of a tradeoff among the number of cells, the size of a cell, and the search time. For example, suppose we are searching among n points in the d dimensional cube, we use poly(n, d) cells, each containing poly(d, log u) bits. We get a lower bound of Ω(d/log n) on the search time, a significant improvement over the recent bound of Ω(log d) of Borodin et al. This should be contrasted with the upper bound of O(log log d) for approximate search (and O(1) for a decision version of the problem; our lower bounds hold in that case). By previous results, the bounds for the cube imply similar bounds for nearest neighbor search in high dimensional Euclidean space, and for other geometric problems.
Bibliographical noteFunding Information:
1Work partially supported by the Milton and Lillian Edwards Fellowship. 2This work was supported by Grant 386/99-1 of the Israel Science Foundation founded by the Israeli Academy of Sciences and Humanities, by the N. Haar and R. Zinn Research Fund, and by the Fund for the Promotion of Research at the Technion.