Bounds on Generalized Huffman Codes

New lower and upper bounds are obtained for the compression of optimal binary prefix codes according to various nonlinear codeword length objectives. Like the coding bounds for Huffman coding — which concern the traditional linear code objective of minimizing average codeword length — these are in terms of a form of entropy and the probability of the most probable input symbol. As in Huffman coding, some upper bounds can be found using sufficient conditions for the codeword corresponding to the most probable symbol being one bit long. Whereas having probability no less than 0.4 is a tight sufficient condition for this to be the case in Huffman coding, other penalties differ, some having a tighter condition, some a looser condition, and others having no such sufficient condition. The objectives explored here are ones for which optimal codes can be found using a generalized form of Huffman coding. These objectives include one related to queueing (an increasing exponential average), one related to single-shot communications (a decaying exponential average), and the recently formulated minimum maximum pointwise redundancy. For these three objectives, we also investigate the necessary and sufficient conditions for the existence of an optimal code with a 1-bit codeword. In the last of them — and in a related fourth objective, dth exponential redundancy — multiple bounds are obtained for different intervals, much as in traditional Huffman coding. Index Terms Huffman codes, minimax redundancy, optimal prefix code, queueing, Rényi entropy.