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

Efficient Construction of Neighborhood Graphs by the Multiple Sorting Method