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A Novel Representation for Permutations
IEEE Transactions on Information Theory ( IF 2.5 ) Pub Date : 2021-01-05 , DOI: 10.1109/tit.2020.3048905
Won-Ho Ri 1 , Ok-Hyon Song 1
Affiliation  

In this paper, we present a variable-length binary code for permutations of degree n , where n is a power of 2. The Lehmer code and its variants provide a bijection between permutations and the indexes in lexicographic ordering. They provide compression of permutations approaching Shannon bound at the expense of structural information. The variable-length code proposed in this paper has asymptotically optimal compression capability while preserving structural information of permutations. In other words, the code encodes the way a permutation is organized, and is optimal in the sense that the ratio of average codeword length and the entropy of permutations approaches 1 as n tends to infinity. The encoding and decoding are efficient. Like the Lehmer code and other enumerative codes, it is not necessary to construct look-up tables. The code is complete and satisfies the prefix condition. Furthermore, the codeword length indicates how “well shuffled” a permutation is. Permutations with longer codewords seem more “random.” An enumeration method of the variable-length code is also given by using T. Cover’s enumerative coding scheme and Schalkwijk code. This gives a different indexing from the Lehmer code. This work can be extended to the case of arbitrary $n$ in a straightforward way.

中文翻译:

排列的新颖表示

在本文中,我们提出了一个可变长度的二进制代码,用于排列度 ñ , 在哪里 ñ是2的幂。Lehmer代码及其变体在词典顺序的排列和索引之间提供了双射。它们以结构信息为代价提供接近香农边界的置换压缩。本文提出的变长码在保持置换结构信息的同时,具有渐近最优的压缩能力。换句话说,代码对排列的组织方式进行编码,并且在平均码字长度与排列熵的比率接近1的意义上说是最优的ñ趋于无穷大。编码和解码是有效的。像Lehmer代码和其他枚举代码一样,不必构造查找表。代码已完成,并且满足前缀条件。此外,码字长度指示排列的“重组程度”。较长代码字的排列似乎更“随机”。还使用T.Cover的枚举编码方案和Schalkwijk码给出了变长码的枚举方法。这给出了与Lehmer代码不同的索引。这项工作可以扩展到任意情况 $ n $ 以一种直接的方式。
更新日期:2021-02-19
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