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Open AccessArticle

A Catalog of Self-Affine Hierarchical Entropy Functions

Department of Electrical & Computer Engineering, University of Minnesota Twin Cities, 200 Union Street SE, Minneapolis, MN 55455, USA
Algorithms 2011, 4(4), 307-333;
Received: 23 September 2011 / Revised: 18 October 2011 / Accepted: 30 October 2011 / Published: 1 November 2011
(This article belongs to the Special Issue Data Compression, Communication and Processing)
PDF [320 KB, uploaded 4 November 2011]


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. View Full-Text
Keywords: types; type classes; lossless compression; hierarchical entropy; self-affine functions; iterated function systems types; type classes; lossless compression; hierarchical entropy; self-affine functions; iterated function systems
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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

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