Search trees are commonly used to implement access operations to a set of stored keys. If this set is static and the probabilities of membership queries are known in advance, then one can precompute an optimal search tree, namely one that minimizes the expected access cost. For a non-key query, a search tree can determine its approximate location by returning the inter-key interval containing the query. This is in contrast to other dictionary data structures, like hash tables, that only report a failed search. We address the question ``what is the additional cost of determining approximate locations for non-key queries''? We prove that for two-way comparison trees this additional cost is at most 1. Our proof is based on a novel probabilistic argument that involves converting a search tree that does not identify non-key queries into a random tree that does.

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基于双向比较的搜索树中搜索失败的代价

搜索树通常用于对一组存储的密钥实施访问操作。如果该集合是静态的，并且成员资格查询的概率是事先已知的，则可以预先计算最佳搜索树，即使期望访问成本最小化的搜索树。对于非关键查询，搜索树可以通过返回包含查询的关键间隔来确定其大概位置。这与仅报告搜索失败的其他字典数据结构（例如哈希表）形成对比。我们解决的问题是``确定非关键查询的大概位置的额外费用是多少？'' 我们证明，对于双向比较树，此额外成本最多为1。