Integer Coding with Nonlinear Costs

Let P = {p(i)} be a measure of strictly positive probabilities on the set of nonnegative integers. Although the countable number of inputs prevents usage of the Huffman algorithm, there are nontrivial P for which known methods find a source code that is optimal in the sense of minimizing expected codeword length. For some applications, however, a source code should instead minimize one of a family of nonlinear objective functions, β-exponential means, those of the form loga ∑ i p(i)a, where n(i) is the length of the ith codeword and a is a positive constant. Applications of such minimizations include minimizing the chance of buffer overflow in a queueing system. This paper introduces algorithms for finding integer codes optimal for such exponential means. One algorithm applies to geometric distributions, while another applies to distributions with lighter tails. The latter algorithm is applied to Poisson distributions and both are extended to alphabetic codes, as well as to minimizing maximum pointwise redundancy. The aforementioned application of minimizing the chance of buffer overflow is also considered.