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In this paper, we address clustering problems in scenarios where accurate direct access to the full dataset is impractical or impossible. Instead, we leverage oracle-based methods, which are particularly valuable in real-world applications where the data may be noisy, restricted due to privacy concerns or sheer volume. We utilize two oracles, the quadruplet and the distance oracle. The quadruplet oracle is a weaker oracle that only approximately compares the distances of two pairs of vertices. In practice, these oracles can be implemented using crowdsourcing or training classifiers or other predictive models. On the other hand, the distance oracle returns exactly the distance of two vertices, so it is a stronger and more expensive oracle to implement. We consider two noise models for the quadruplet oracle. In the adversarial noise model, if two pairs have similar distances, the response is chosen by an adversary. In the probabilistic noise model, the pair with the smaller distance is returned with a constant probability. We consider a set V of n vertices in a metric space that supports the quadruplet and the distance oracle. For each of the k-center, k-median, and k-means clustering problem on V, we design constant approximation algorithms that perform roughly O(nk) calls to the quadruplet oracle and O(k^2) calls to the distance oracle in both noise models. When the dataset has low intrinsic dimension, we significantly improve the approximation factors of our algorithms by performing a few additional calls to the distance oracle. We also show that for k-median and k-means clustering there is no hope to return any sublinear approximation using only the quadruplet oracle. Finally, we give constant approximation algorithms for estimating the clustering cost induced by any set of k vertices, performing roughly O(nk) calls to the quadruplet oracle and O(k^2) calls to the distance oracle. 

We are given a set 𝒵 = {(R_1,s_1), …, (R_n,s_n)}, where each R_i is a range in ℝ^d, such as rectangle or ball, and s_i ∈ [0,1] denotes its selectivity. The goal is to compute a small-size discrete data distribution 𝒟 = {(q₁,w₁),…, (q_m,w_m)}, where q_j ∈ ℝ^d and w_j ∈ [0,1] for each 1 ≤ j ≤ m, and ∑_{1≤j≤m} w_j = 1, such that 𝒟 is the most consistent with 𝒵, i.e., err_p(𝒟,𝒵) = 1/n ∑_{i = 1}ⁿ |s_i - ∑_{j=1}^m w_j⋅1(q_j ∈ R_i)|^p is minimized. In a database setting, 𝒵 corresponds to a workload of range queries over some table, together with their observed selectivities (i.e., fraction of tuples returned), and 𝒟 can be used as compact model for approximating the data distribution within the table without accessing the underlying contents. In this paper, we obtain both upper and lower bounds for this problem. In particular, we show that the problem of finding the best data distribution from selectivity queries is NP-complete. On the positive side, we describe a Monte Carlo algorithm that constructs, in time O((n+δ^{-d}) δ^{-2} polylog n), a discrete distribution 𝒟̃ of size O(δ^{-2}), such that err_p(𝒟̃,𝒵) ≤ min_𝒟 err_p(𝒟,𝒵)+δ (for p = 1,2,∞) where the minimum is taken over all discrete distributions. We also establish conditional lower bounds, which strongly indicate the infeasibility of relative approximations as well as removal of the exponential dependency on the dimension for additive approximations. This suggests that significant improvements to our algorithm are unlikely.