Best Huffman trees

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Acta Informatica, ISSN: 0001-5903, Vol: 16, Issue: 3, Page: 363-370

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Markowsky, George
Springer Nature America, Inc; Springer Verlag
Computer Science; Computer Sciences
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Given a sequence of positive weights, W=w≧...≧w>0, there is a Huffman tree, T↑ ("T-up") which minimizes the following functions: max{d(w)}; Σd(w); Σf(d(w)) w(here d(w) represents the distance of a leaf of weight w to the root and f is a function defined for nonnegative integers having the property that g(x) = f(x + 1) - f(x) is monotone increasing) over the set of all trees for W having minimal expected length. Minimizing the first two functions was first done by Schwartz [5]. In the case of codes where W is a sequence of probabilities, this implies that the codes based on T↑ have all their absolute central moments minimal. In particular, they are the least variance codes which were also described by Kou [3]. Furthermore, there exists a Huffman tree T↓, ("T-down") which maximizes the functions considered above. However, if g(x) is monotone decreasing, T↑ and T↓, respectively maximize and minimize Σf(d(w) w) over the set of all trees for W having minimal expected length. In addition, we derive a number of interesting results about the distribution of labels within Huffman trees. By suitable modifications of the usual Huffman tree construction, (see [1]) T↑ and T↓ can also be constructed in time O(n log n). © 1981 Springer-Verlag.