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Similarity search is a fundamental algorithmic primitive, widely used in many computer science disciplines. Given a set of points S and a radius parameter r > 0, the r -near neighbor ( r -NN) problem asks for a data structure that, given any query point q , returns a point p within distance at most r from q. In this paper, we study the r -NN problem in the light of individual fairness and providing equal opportunities: all points that are within distance r from the query should have the same probability to be returned. The problem is of special interest in high dimensions, where Locality Sensitive Hashing (LSH), the theoretically leading approach to similarity search, does not provide any fairness guarantee. In this work, we show that LSH-based algorithms can be made fair, without a significant loss in efficiency. We propose several efficient data structures for the exact and approximate variants of the fair NN problem. Our approach works more generally for sampling uniformly from a sub-collection of sets of a given collection and can be used in a few other applications. We also carried out an experimental evaluation that highlights the inherent unfairness of existing NN data structures. 

In this paper, we prove a two-sided variant of the Kirszbraun theorem. Consider an arbitrary subset X of Euclidean space and its superset Y. Let f be a 1-Lipschitz map from X to ℝ^m. The Kirszbraun theorem states that the map f can be extended to a 1-Lipschitz map ̃ f from Y to ℝ^m. While the extension ̃ f does not increase distances between points, there is no guarantee that it does not decrease distances significantly. In fact, ̃ f may even map distinct points to the same point (that is, it can infinitely decrease some distances). However, we prove that there exists a (1 + ε)-Lipschitz outer extension f̃:Y → ℝ^{m'} that does not decrease distances more than "necessary". Namely, ‖f̃(x) - f̃(y)‖ ≥ c √{ε} min(‖x-y‖, inf_{a,b ∈ X} (‖x - a‖ + ‖f(a) - f(b)‖ + ‖b-y‖)) for some absolutely constant c > 0. This bound is asymptotically optimal, since no L-Lipschitz extension g can have ‖g(x) - g(y)‖ > L min(‖x-y‖, inf_{a,b ∈ X} (‖x - a‖ + ‖f(a) - f(b)‖ + ‖b-y‖)) even for a single pair of points x and y. In some applications, one is interested in the distances ‖f̃(x) - f̃(y)‖ between images of points x,y ∈ Y rather than in the map f̃ itself. The standard Kirszbraun theorem does not provide any method of computing these distances without computing the entire map ̃ f first. In contrast, our theorem provides a simple approximate formula for distances ‖f̃(x) - f̃(y)‖. 

We introduce and study the notion of *an outer bi-Lipschitz extension* of a map between Euclidean spaces. The notion is a natural analogue of the notion of *a Lipschitz extension* of a Lipschitz map. We show that for every map f there exists an outer bi-Lipschitz extension f′ whose distortion is greater than that of f by at most a constant factor. This result can be seen as a counterpart of the classic Kirszbraun theorem for outer bi-Lipschitz extensions. We also study outer bi-Lipschitz extensions of near-isometric maps and show upper and lower bounds for them. Then, we present applications of our results to prioritized and terminal dimension reduction problems, described next. We prove a *prioritized* variant of the Johnson–Lindenstrauss lemma: given a set of points X⊂ ℝd of size N and a permutation (”priority ranking”) of X, there exists an embedding f of X into ℝO(logN) with distortion O(loglogN) such that the point of rank j has only O(log3 + ε j) non-zero coordinates – more specifically, all but the first O(log3+ε j) coordinates are equal to 0; the distortion of f restricted to the first j points (according to the ranking) is at most O(loglogj). The result makes a progress towards answering an open question by Elkin, Filtser, and Neiman about prioritized dimension reductions. We prove that given a set X of N points in ℜd, there exists a *terminal* dimension reduction embedding of ℝd into ℝd′, where d′ = O(logN/ε4), which preserves distances ||x−y|| between points x∈ X and y ∈ ℝd, up to a multiplicative factor of 1 ± ε. This improves a recent result by Elkin, Filtser, and Neiman. The dimension reductions that we obtain are nonlinear, and this nonlinearity is necessary. 

We study the classic set cover problem from the perspective of sub-linear algorithms. Given access to a collection of m sets over n elements in the query model, we show that sub-linear algorithms derived from existing techniques have almost tight query complexities. On one hand, first we show an adaptation of the streaming algorithm presented in [17] to the sub-linear query model, that returns an α-approximate cover using Õ(m(n/k)^1/(α–1) + nk) queries to the input, where k denotes the value of a minimum set cover. We then complement this upper bound by proving that for lower values of k, the required number of queries is , even for estimating the optimal cover size. Moreover, we prove that even checking whether a given collection of sets covers all the elements would require Ω(nk) queries. These two lower bounds provide strong evidence that the upper bound is almost tight for certain values of the parameter k. On the other hand, we show that this bound is not optimal for larger values of the parameter k, as there exists a (1 + ε)-approximation algorithm with Õ(mn/kε^2) queries. We show that this bound is essentially tight for sufficiently small constant ε, by establishing a lower bound of query complexity. Our lower-bound results follow by carefully designing two distributions of instances that are hard to distinguish. In particular, our first lower bound involves a probabilistic construction of a certain set system with a minimum set cover of size αk, with the key property that a small number of “almost uniformly distributed” modifications can reduce the minimum set cover size down to k. Thus, these modifications are not detectable unless a large number of queries are asked. We believe that our probabilistic construction technique might find applications to lower bounds for other combinatorial optimization problems.