Minimal Huffman trees

Abstract.

For a given set of real weights, Huffman trees minimize the weighted external path length. Over the years, several algorithms have been proposed for constructing Huffman trees that minimize additional natural cost functions such as the external path length, the variance and, more generally, the central moments. We show that all these cost functions are minimized on exactly the same class of Huffman trees, and we characterize the class: it consists of all Huffman trees of minimal level set. It follows that a Huffman tree minimizing one of the cost functions in fact minimizes all of them, and has the minimal level set; in particular, it has minimum height. We show that the unique Huffman tree produced by the simplest construction method, the bottom-merge algorithm of Schwartz, belongs to the class. Finally, we prove that several natural variants of Huffman's algorithm, that appear to be nondeterministic, in fact all lead to the single Huffman tree obtained by Schwartz's algorithm.