Centrality indices, such as closeness and eccentricity, are key to identifying influential nodes within a network, with applications ranging from social and biological networks to communication and transportation systems. However, computing these indices for every node in large graphs is computationally prohibitive due to the need for solving the All-Pairs Shortest Path (APSP) problem. This paper introduces a framework for approximating closeness and eccentricity centrality by selecting a sequence of strategically chosen anchor nodes, from which Breadth-First Searches (BFS) are performed. We present two anchor-selection strategies that minimize estimation error for these indices and evaluate their effectiveness on synthetic and real-world networks. Comparative results indicate that while random anchor selection occasionally achieves lower errors for closeness, other strategies outperform in eccentricity estimation. This study highlights the effectiveness of anchor-based approximations and the trade-offs between different selection methods in estimating centrality at scale.