We consider the problem of learning the true ordering of a set of alternatives from largely incomplete and noisy rankings. We introduce a natural generalization of both the classical Mallows model of ranking distributions and the extensively studied model of noisy pairwise comparisons. Our selective Mallows model outputs a noisy ranking on any given subset of alternatives, based on an underlying Mallows distribution. Assuming a sequence of subsets where each pair of alternatives appears

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... ly enough, we obtain strong asymptotically tight upper and lower bounds on the sample complexity of learning the underlying complete ranking and the (identities and the) ranking of the top-k alternatives from selective Mallows rankings. Moreover, building on the work of (Braverman and Mossel, 2009), we show how to efficiently compute the maximum likelihood complete ranking from selective Mallows rankings.