Source Coding for Quasiarithmetic Penalties

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 but rather a generalized mean known as a quasiarithmetic or quasilinear mean. Such generalized 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.Comment: 22 pages, 3 figures, submitted to IEEE Trans. Inform. Theory, revised per suggestions of reader

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

Lossless quantum data compression with exponential penalization: an operational interpretation of the quantum Rényi entropy

Based on the problem of quantum data compression in a lossless way, we present here an operational interpretation for the family of quantum Rényi entropies. In order to do this, we appeal to a very general quantum encoding scheme that satisfies a quantum version of the Kraft-McMillan inequality. Then, in the standard situation, where one is intended to minimize the usual average length of the quantum codewords, we recover the known results, namely that the von Neumann entropy of the source bounds the average length of the optimal codes. Otherwise, we show that by invoking an exponential average length, related to an exponential penalization over large codewords, the quantum Rényi entropies arise as the natural quantities relating the optimal encoding schemes with the source description, playing an analogous role to that of von Neumann entropy.Facultad de Ciencias Exacta

Generalizations of Length Limited Huffman Coding for Hierarchical Memory Settings

In this paper, we study the problem of designing prefix-free encoding schemes having minimum average code length that can be decoded efficiently under a decode cost model that captures memory hierarchy induced cost functions. We also study a special case of this problem that is closely related to the length limited Huffman coding (LLHC) problem; we call this the soft-length limited Huffman coding problem. In this version, there is a penalty associated with each of the n characters of the alphabet whose encodings exceed a specified bound D(? n) where the penalty increases linearly with the length of the encoding beyond D. The goal of the problem is to find a prefix-free encoding having minimum average code length and total penalty within a pre-specified bound P. This generalizes the LLHC problem. We present an algorithm to solve this problem that runs in time O(nD). We study a further generalization in which the penalty function and the objective function can both be arbitrary monotonically non-decreasing functions of the codeword length. We provide dynamic programming based exact and PTAS algorithms for this setting