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Huffman coding finds a prefix code that minimizes mean codeword length for a given probability distribution over a finite number of items. Campbell generalized the Huffman problem to a family of problems in which the goal is to minimize not mean codeword length i pili but rather a generalized mean of the form ϕ −1 ( i piϕ(li)),where li denotes the length of the ith codeword, pi denotes the corresponding probability, and ϕ is a monotonically increasing cost function. Such generalized means -also known as quasiarithmetic or quasilinear means -have a number of diverse applications, including applications in queueing. Several quasiarithmetic-mean problems have novel simple redundancy bounds in terms of a generalized entropy. A related property involves the existence of optimal codes: For "well-behaved" cost functions, optimal codes always exist for (possibly infinite-alphabet) sources having finite generalized entropy. Solving finite instances of such problems is done by generalizing an algorithm for finding length-limited binary codes to a new algorithm for finding optimal binary codes for any quasiarithmetic mean with a convex cost function. This algorithm can be performed using quadratic time and linear space, and can be extended to other penalty functions, some of which are solvable with similar space and time complexity, and others of which are solvable with slightly greater complexity. This reduces the computational complexity of a problem involving minimum delay in a queue, allows combinations of previously considered problems to be optimized, and greatly expands the space of problems solvable in quadratic time and linear space. The algorithm can be extended for purposes such as breaking ties among possibly different optimal codes, as with bottom-merge Huffman coding. Index TermsOptimal prefix code, Huffman algorithm, generalized entropies, generalized means, quasiarithmetic means, queueing.

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.

A framework with two scalar parameters is introduced for various problems of finding a prefix code minimizing a coding penalty function. The framework encompasses problems previously proposed by Huffman, Campbell, Nath, and Drmota and Szpankowski, shedding light on the relationships among these problems. In particular, Nath's range of problems can be seen as bridging the minimum average redundancy problem of Huffman with the minimum maximum pointwise redundancy problem of Drmota and Szpankowski. Using this framework, two linear-time Huffman-like algorithms are devised for the minimum maximum pointwise redundancy problem, the only one in the framework not previously solved with a Huffman-like algorithm. Both algorithms provide solutions common to this problem and a subrange of Nath's problems, the second algorithm being distinguished by its ability to find the minimum variance solution among all solutions common to the minimum maximum pointwise redundancy and Nath problems. Simple redundancy bounds are also presented.

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