This article establishes the kn −k(k+1)2 + 1/kn −k(k+1)2 + k − 1-diagnosability of n-dimensional hierarchical star graphs HSn (n ≥ 4) underthe PMC model and the MM* model, where 2 ≤ k ≤ n − 2. Moreover, novelt/s diagnostic algorithms are proposed for the rapid localization of faulty processors within both the PMC model and the MM* model. These algorithms exhibita time complexity of O(N log2 N), where N represents the total number ofprocessors in the system.