Improved Algorithms for Clustering with Noisy Distance Oracles

Bateni *et al.* has recently introduced the *weak-strong distance oracle model* to study clustering problems in settings with limited distance information. Given query access to the strong-oracle and weak-oracle in the weak-strong oracle model, the authors design approximation algorithms for $k$-means and $k$-center clustering problems. In this work, we design algorithms with improved guarantees for $k$-means and $k$-center clustering problems in the weak-strong oracle model. The $k$-means++ algorithm is routinely used to solve $k$-means in settings where complete distance information is available. One of the main contributions of this work is to show that $k$-means++ algorithm can be adapted to work in the weak-strong oracle model using only a small number of strong-oracle queries, which is the critical resource in this model. In particular, our $k$-means++ based algorithm gives a constant approximation for $k$-means and uses $O(k^2 \log^2{n})$ strong-oracle queries. This improves on the algorithm of Bateni *et al.* that uses $O(k^2 \log^4n \log^2 \log n)$ strong-oracle queries for a constant factor approximation of $k$-means. For the $k$-center problem, we give a simple *ball-carving* based $6(1 + \epsilon)$-approximation algorithm that uses $O(k^3 \log^2{n} \log{\frac{\log{n}}{\epsilon}})$ strong-oracle queries. This is an improvement over the $14(1 + \epsilon)$-approximation algorithm of Bateni *et al.* that uses $O(k^2 \log^4{n} \log^2{\frac{\log{n}}{\epsilon}})$ strong-oracle queries. To show the effectiveness of our algorithms, we perform empirical evaluations on real-world datasets and show that our algorithms significantly outperform the algorithms of Bateni *et al.*