Better algorithms for high-dimensional proximity problems via asymmetric embeddings

In this paper we give several results based on randomized embeddings of <i>l</i><inf>2</inf> into <i>l</i><inf>∞</inf>(or "<i>l</i><inf>∞</inf>-like") spaces. Our first result is a (1 + ε)-distortion <i>asymmetric</i> embedding of <i>n</i> points in <i>l</i><inf>2</inf> into <i>l</i><inf>∞</inf> with polylog(<i>n</i>) dimension, for any 1 + ε. This gives the first known <i>O</i>(1)- approximate nearest neighbor algorithm with fast query time and almost polynomial space for a product of Euclidean norms, a common generalization of both <i>l</i><inf>2</inf> and <i>l</i><inf>∞</inf> norms. Our embedding also clarifies the relative complexity of approximate nearest neighbor in <i>l</i><inf>2</inf> and <i>l</i><inf>∞</inf> spaces.Our second result in a (1 + ε)-approximate algorithm for the diameter of <i>n</i> points in <i>l<sup>d</sup><inf>2</inf></i>, running in time <i>Õ</i>(<i>dn</i><sup>1+l/(1+ε)</sup><sup>2</sup>); the algorithm is fully dynamic. This improves several previous algorithms for this problem (see Table 1 for more information).

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