Gagie and Nekrich (2009) gave an algorithm for adaptive prefix-free coding that, given a string $S [1..n]$ over the alphabet $\{1, \ldots, \sigma\}$ with $\sigma = o (n / \log^{5 / 2} n)$, encodes $S$ in at most $n (H + 1) + o (n)$ bits, where $H$ is the empirical entropy of $S$, such that encoding and decoding $S$ take $O (n)$ time. They also proved their bound on the encoding length is optimal, even when the empirical entropy is high. Their algorithm is impractical, however, because it uses complicated data structures. In this paper we give an algorithm with the same bounds, except that we require $\sigma = o (n^{1 / 2} / \log n)$, that uses no data structures more complicated than a lookup table. Moreover, when Gagie and Nekrich's algorithm is used for optimal adaptive alphabetic coding it takes $O (n \log \log n)$ time for decoding, but ours still takes $O (n)$ time.

中文翻译：

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简单的最坏情况最优自适应无前缀编码

Gagie 和 Nekrich (2009) 给出了一种自适应无前缀编码算法，给定一个字符串 $S [1..n]$ 在字母表 $\{1, \ldots, \sigma\}$ 中，$\sigma = o (n / \log^{5 / 2} n)$, 将 $S$ 编码到最多 $n (H + 1) + o (n)$ 位，其中 $H$ 是 $S$ 的经验熵，这样编码和解码 $S$ 花费 $O (n)$ 时间。他们还证明了他们对编码长度的限制是最优的，即使经验熵很高。然而，他们的算法是不切实际的，因为它使用了复杂的数据结构。在本文中，我们给出了一个具有相同边界的算法，只是我们需要 $\sigma = o (n^{1 / 2} / \log n)$，该算法不使用比查找表更复杂的数据结构。此外，当 Gagie 和 Nekrich 的算法用于最优自适应字母编码时，解码需要 $O (n \log \log n)$ 时间，