Lower Bounds on Near Neighbor Search via Metric Expansion

In this paper we show how the complexity of performing nearest neighbor (NNS) search on a metric space is related to the expansion of the metric space. Given a metric space we look at the graph obtained by connecting every pair of points within a certain distance $r$ . We then look at various notions of expansion in this graph relating them to the cell probe complexity of NNS for randomized and deterministic, exact and approximate algorithms. For example if the graph has node expansion $\Phi$ then we show that any deterministic $t$-probe data structure for $n$ points must use space $S$ where $(St/n)^t > \Phi$. We show similar results for randomized algorithms as well. These relationships can be used to derive most of the known lower bounds in the well known metric spaces such as $l_1$, $l_2$, $l_\infty$, and some new ones, by simply computing their expansion. In the process, we strengthen and generalize our previous results~\cite{PTW08}. Additionally, we unify the approach in~\cite{PTW08} and the communication complexity based approach. Our work reduces the problem of proving cell probe lower bounds of near neighbor search to computing the appropriate expansion parameter. In our results, as in all previous results, the dependence on $t$ is weak, that is, the bound drops exponentially in $t$. We show a much stronger (tight) time-space tradeoff for the class of \emph{dynamic} \emph{low contention} data structures. These are data structures that supports updates in the data set and that do not look up any single cell too often. A full version of the paper could be found in [19].