Optimal Prefix Codes for Infinite Alphabets With Nonlinear Costs

Abstract

Abstract-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 log a P i p(i)a n(i) , where n(i) is the length of the ith 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.

Lossless coding with generalized criteria

“…According to (6), (7), for y / ∈ U the weight w α (y) decreases, and for x ∈ U the weight w α (x) increases. Hence, since w α (·) is a continuous function with respect to α, at some α = α , w α (x) = w α (y) = w * α .…”

“…Suppose that for some α = α + dα, dα > 0, w α (x) = w α (y). Then, the largest weight will decrease and the lowest weight will increase as a function of α ∈ [0, 1] according to (6) and (7), respectively. Remark 1.…”

“…which is precisely the solution of a variant of the Shannon code, minimizing the average of an exponential function of the codeword length pay-off [5], [6]. …”

Rank Modulation for Flash Memories

“…Now let us assume we have a cyclic and complete -RMGC, which we call , defined by the sequence of transitions and where , i.e., a "push-to-the-top" operation on the second element in the permutation. 1 We further assume that the transition appears at least twice. We will now show how to construct , a cyclic and complete -RMGC with the same property.…”

“…In [20], the Huffman code construction was generalized, assuming that the vertex-degree distribution in the code tree is given. In [1], prefix-free codes for infinite alphabets and nonlinear costs were presented. When the letters of the encoding alphabet have unequal lengths, only exponential-time algorithms are known, and it is not known yet whether this problem is NP-hard [12].…”