We consider the learning and communication complexity of subsequence containment. In the learning problem, we seek to learn a classifier that positively labels a binary string $x$ if it contains a fixed binary string $y$ as a subsequence. In the communication problem, $x$ and $y$ are partitioned between two players, Alice and Bob, who wish to determine if $x$ contains $y$ as a subsequence using a minimal amount of communication. We devise asymptotically tight bounds for the sample complexity (VC dimension) of the learning problem and the communication complexity of the communication problem. Our results illustrate that the sample complexity of our learning problem can be considerably larger when the subsequence occurs in non-contiguous locations.

翻译：我们考虑了后序列封隔的学习和通信复杂性。 在学习问题中,我们试图学习一个精细的分类师,如果其中含有固定的二进制字符串,则该分类师对二进制字符串($x$x)贴上肯定的标签。在通信问题中,美元和美元由两个玩家(Alice和Bob)分割,他们希望用最低限度的通信量来确定美元是否包含以美元作为子序列。我们为学习问题和通信问题通信复杂性的样本复杂性(VC维度)设计了不那么紧凑的界限。我们的结果表明,当子序列发生在不相干的地方时,我们学习问题的样本复杂性会大得多。