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Skeleton Learning (SL) is the task for learning an undirected graph from the input data that captures their dependency relations. SL plays a pivotal role in causal learning and has attracted growing attention in the research community lately. Due to the high time complexity, anytime SL has emerged which learns a skeleton incrementally and improves it overtime. In this paper, we first propose and advocate the reliability requirement for anytime SL to be practically useful. Reliability requires the intermediately learned skeleton to have precision and persistency. We also present REAL, a novel Reliable and Efficient Anytime Learning algorithm of skeleton. Specifically, we point out that the commonly existing Functional Dependency (FD) among variables could make the learned skeleton violate faithfulness assumption, thus we propose a theory to resolve such incompatibility. Based on this, REAL conducts SL on a reduced set of variables with guaranteed correctness thus drastically improves efficiency. Furthermore, it employs a novel edge-insertion and best-first strategy in anytime fashion for skeleton growing to achieve high reliability and efficiency. We prove that the skeleton learned by REAL converges to the correct skeleton under standard assumptions. Thorough experiments were conducted on both benchmark and real-world datasets demonstrate that REAL significantly outperforms the other state-of-the-art algorithms.