The Renyi redundancy of generalized Huffman codes

Huffman´s algorithm gives optimal codes, as measured by average codeword length, and the redundancy can be measured as the difference between the average codeword length and Shannon´s entropy. If the objective function is replaced by an exponentially weighted average, then a simple modification of Huffman´s algorithm gives optimal codes. The redundancy can now be measured as the difference between this new average and A. Renyi´s (1961) generalization of Shannon´s entropy. By decreasing some of the codeword lengths in a Shannon code, the upper bound on the redundancy given in the standard proof of the noiseless source coding theorem is improved. The lower bound is improved by randomizing between codeword lengths, allowing linear programming techniques to be used on an integer programming problem. These bounds are shown to be asymptotically equal. The results are generalized to the Renyi case and are related to R.G. Gallager´s (1978) bound on the redundancy of Huffman codes