On the redundancy of Huffman codes with exponential objectives

We present new lower and upper bounds for the compression rate of binary prefix codes over memoryless sources optimized according to two related exponential codeword length objectives. The objectives explored here are exponential-average length and exponential-average redundancy. The first of these relates to various problems involving queueing, uncertainty, and lossless communications. It can be reduced to the second, which has properties more amenable to analysis. These bounds, some of which are tight, are in terms of a form of entropy and/or the probability of an input symbol, improving on recently discovered bounds of similar form. We also observe related properties of optimal codes over the exponential-average redundancy utility.