Optimal Prefix Codes for Infinite Alphabets 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, $\beta$-exponential means, those of the form $\log_a \sum_i p(i) a^{n(i)}$, where $n(i)$ is the length of the $i$th codeword and $a$ is a positive constant. Applications of such minimizations include a novel problem of maximizing the chance of message receipt in single-shot communications ($a<1$) and a previously known problem of minimizing the chance of buffer overflow in a queueing system ($a>1$). This paper introduces methods for finding codes optimal for such exponential means. One method 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.Comment: 14 pages, 6 figures, accepted to IEEE Trans. Inform. Theor

The Victims of NIMBY

Not In My Back Yard, or NIMBY, in its various forms, has three principal types of targets. The first is waste disposal facilities, primarily landfills and incinerators. The second is low-income housing. The third is social service facilities, group homes and shelters for individuals such as the mentally ill, AIDS patients, and the homeless. This Article addresses the issue of the victims of NIMBY, with special reference to the effects of project opposition on racial minorities. Because the effect of facility opposition varies widely with the type of project involved, Part II arrays the types of relevant projects and shows the ways that opposition manifests itself. Part III then briefly discusses the legal techniques used by those who oppose facilities, and the counter measures used by facility proponents. Part IV examines the available evidence on who suffers as a result of the opponents\u27 techniques. Part V looks at who benefits from opposition to siting new facilities. Part VI is devoted to some of the secondary and imponderable effects of facility opposition. Finally, Part VII draws conclusions from the preceding discussion, and shows how the costs and benefits of NIMBY are very different from those envisioned by those who either condemn or applaud facility opposition