Algorithms 2011, 4(4), 307-333; doi:10.3390/a4040307
Article

A Catalog of Self-Affine Hierarchical Entropy Functions

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Received: 23 September 2011; in revised form: 18 October 2011 / Accepted: 30 October 2011 / Published: 1 November 2011
(This article belongs to the Special Issue Data Compression, Communication and Processing)
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract: For fixed k ≥ 2 and fixed data alphabet of cardinality m, the hierarchical type class of a data string of length n = kj for some j ≥ 1 is formed by permuting the string in all possible ways under permutations arising from the isomorphisms of the unique finite rooted tree of depth j which has n leaves and k children for each non-leaf vertex. Suppose the data strings in a hierarchical type class are losslessly encoded via binary codewords of minimal length. A hierarchical entropy function is a function on the set of m-dimensional probability distributions which describes the asymptotic compression rate performance of this lossless encoding scheme as the data length n is allowed to grow without bound. We determine infinitely many hierarchical entropy functions which are each self-affine. For each such function, an explicit iterated function system is found such that the graph of the function is the attractor of the system.
Keywords: types; type classes; lossless compression; hierarchical entropy; self-affine functions; iterated function systems
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MDPI and ACS Style

Kieffer, J. A Catalog of Self-Affine Hierarchical Entropy Functions. Algorithms 2011, 4, 307-333.

AMA Style

Kieffer J. A Catalog of Self-Affine Hierarchical Entropy Functions. Algorithms. 2011; 4(4):307-333.

Chicago/Turabian Style

Kieffer, John. 2011. "A Catalog of Self-Affine Hierarchical Entropy Functions." Algorithms 4, no. 4: 307-333.

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