An optimal algorithm for approximate nearest neighbor searching fixed dimensions

Consider a set of <italic>S</italic> of <italic>n</italic> data points in real <italic>d</italic>-dimensional space, R<supscrpt>d</supscrpt>, where distances are measured using any Minkowski metric. In nearest neighbor searching, we preprocess <italic>S</italic> into a data structure, so that given any query point <italic>q</italic><inline-equation> <f>∈</f></inline-equation> R<supscrpt>d</supscrpt>, is the closest point of S to <italic>q</italic> can be reported quickly. Given any positive real ε, data point <italic>p</italic> is a (1 +ε)-<italic>approximate nearest neighbor</italic> of <italic>q</italic> if its distance from <italic>q</italic> is within a factor of (1 + ε) of the distance to the true nearest neighbor. We show that it is possible to preprocess a set of <italic>n</italic> points in R<supscrpt>d</supscrpt> in <italic>O(dn</italic> log <italic>n</italic>) time and <italic>O(dn)</italic> space, so that given a query point <italic> q</italic> <inline-equation> <f>∈</f></inline-equation> R<supscrpt>d</supscrpt>, and ε > 0, a (1 + ε)-approximate nearest neighbor of <italic>q</italic> can be computed in <italic>O</italic>(<italic>c</italic><subscrpt><italic>d</italic>, ε</subscrpt> log <italic>n</italic>) time, where <italic>c<subscrpt>d,ε</subscrpt></italic>≤<italic>d</italic> <inline-equation> <f><fen lp="ceil">1 + 6d/<g>e</g><rp post="ceil"></fen></f></inline-equation>;<supscrpt>d</supscrpt> is a factor depending only on dimension and ε. In general, we show that given an integer <italic>k</italic> ≥ 1, (1 + ε)-approximations to the <italic>k</italic> nearest neighbors of <italic>q</italic> can be computed in additional <italic>O(kd</italic> log <italic>n</italic>) time.