  Neighborhood graphs are gaining popularity as a concise data representation
in machine learning. However, naive graph construction by pairwise distance
calculation takes $O(n^2)$ runtime for $n$ data points and this is
prohibitively slow for millions of data points. For strings of equal length,
the multiple sorting method (Uno, 2008) can construct an $\epsilon$-neighbor
graph in $O(n+m)$ time, where $m$ is the number of $\epsilon$-neighbor pairs in
the data. To introduce this remarkably efficient algorithm to continuous
domains such as images, signals and texts, we employ a random projection method
to convert vectors to strings. Theoretical results are presented to elucidate
the trade-off between approximation quality and computation time. Empirical
results show the efficiency of our method in comparison to fast nearest
neighbor alternatives.
