Twenty (or so) Questions: $D$-ary Length-Bounded Prefix Coding
<|reference_start|>Twenty (or so) Questions: $D$-ary Length-Bounded Prefix Coding: Efficient optimal prefix coding has long been accomplished via the Huffman algorithm. However, there is still room for improvement and exploration regarding variants of the Huffman problem. Length-limited Huffman coding, useful for many practical applications, is one such variant, for which codes are restricted to the set of codes in which none of the $n$ codewords is longer than a given length, $l_{\max}$. Binary length-limited coding can be done in $O(n l_{\max})$ time and O(n) space via the widely used Package-Merge algorithm and with even smaller asymptotic complexity using a lesser-known algorithm. In this paper these algorithms are generalized without increasing complexity in order to introduce a minimum codeword length constraint $l_{\min}$, to allow for objective functions other than the minimization of expected codeword length, and to be applicable to both binary and nonbinary codes; nonbinary codes were previously addressed using a slower dynamic programming approach. These extensions have various applications -- including fast decompression and a modified version of the game ``Twenty Questions'' -- and can be used to solve the problem of finding an optimal code with limited fringe, that is, finding the best code among codes with a maximum difference between the longest and shortest codewords. The previously proposed method for solving this problem was nonpolynomial time, whereas solving this using the novel linear-space algorithm requires only $O(n (l_{\max}- l_{\min})^2)$ time, or even less if $l_{\max}- l_{\min}$ is not $O(\log n)$.<|reference_end|>
@article{baer2006twenty,
    title={Twenty (or so) Questions: $D$-ary Length-Bounded Prefix Coding},
    author={Michael B. Baer},
    journal={arXiv preprint arXiv:cs/0602085},
    year={2006},
    archivePrefix={arXiv},
    eprint={cs/0602085},
    primaryClass={cs.IT cs.DS math.IT}
}