Hardness of Nearest Neighbor under L-infinity

Recent years have seen a significant increase in our understanding of high-dimensional nearest neighbor search (NNS) for distances like the lscr₁ and lscr₂ norms. By contrast, our understanding of the lscr_infin norm is now where it was (exactly) 10 years ago. In FOCSpsila98, Indyk proved the following unorthodox result: there is a data structure (in fact, a decision tree) of size O(n^rho), for any rho > 1, which achieves approximation O(log_rho log d) for NNS in the d-dimensional lscr₁ metric. In this paper, we provide results that indicate that Indykpsilas unconventional bound might in fact be optimal. Specifically, we show a lower bound for the asymmetric communication complexity of NNS under lscrinfin, which proves that this space/approximation trade-off is optimal for decision trees and for data structures with constant cell-probe complexity.