On the I/O complexity of the k-nearest neighbor problem
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We consider static, external memory indexes for exact and approximate versions of the k-nearest neighbor (k-NN) problem, and show new lower bounds under a standard indivisibility assumption: - Polynomial space indexing schemes for high-dimensional k-NN in Hamming space cannot take advantage of block transfers: Ω(k) block reads are needed to to answer a query. - For the ℓ_∞ metric the lower bound holds even if we allow c-appoximate nearest neighbors to be returned, for c ∈ (1, 3). - The restriction to c < 3 is necessary: For every metric there exists an indexing scheme in the indexability model of Hellerstein et al. using space O(kn), where n is the number of points, that can retrieve k 3-approximate nearest neighbors using k/B I/Os, which is optimal. - For specific metrics, data structures with better approximation factors are possible. For k-NN in Hamming space and every approximation factor c>1 there exists a polynomial space data structure that returns kc-approximate nearest neighbors in k/B I/Os. To show these lower bounds we develop two new techniques: First, to handle that approximation algorithms have more freedom in deciding which result set to return we develop a relaxed version of the λ-set workload technique of Hellerstein et al. This technique allows us to show lower bounds that hold in d≥ n dimensions. To extend the lower bounds down to d = O(k log(n/k)) dimensions, we develop a new deterministic dimension reduction technique that may be of independent interest.
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