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 lscr
₁
and lscr
₂
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 lscr
₁
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.