We find a searching method on ordered lists that surprisingly outperforms binary searching with respect to average query complexity while retaining minmax optimality. The method is shown to require $O(\log_2\log_2 n)$ queries on average while never exceeding $\lceil \log_2 n \rceil$ queries in the worst case, i.e. the minmax bound of binary searching. Our average results assume a uniform distribution hypothesis similar to those of prevous authors under which the expected query complexity of interpolation search of $O(\log_2\log_2 n)$ is known to be optimal. Hence our method turns out to be optimal with respect to both minmax and average performance. We further provide robustness guarantees and perform several numerical experiments with both artificial and real data. Our results suggest that time savings range roughly from a constant factor of 10\% to 50\% to a logarithmic factor spanning orders of magnitude when different metrics are considered.

中文翻译：

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以$ O（\ log_2 \ log_2 n）$平均成本进行的最小最大最优列表搜索

我们发现一种有序列表上的搜索方法，就平均查询复杂度而言，它在保持minmax最优性的同时，出乎意料地优于二进制搜索。结果表明，该方法平均需要$ O（\ log_2 \ log_2 n）$个查询，而在最坏的情况下（即二进制搜索的minmax范围），则永远不会超过$ \ lceil \ log_2 n \ rceil $个查询。我们的平均结果假设一个均匀分布的假设类似于先前的作者，在该假设下，已知插值搜索$ O（\ log_2 \ log_2 n）$的预期查询复杂度是最佳的。因此，相对于minmax和平均性能，我们的方法被证明是最佳的。我们进一步提供了鲁棒性保证，并使用人工和真实数据进行了一些数值实验。