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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
Received: 23 September 2011; in revised form: 18 October 2011 / Accepted: 30 October 2011 / Published: 1 November 2011
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.
Kieffer J. A Catalog of Self-Affine Hierarchical Entropy Functions. Algorithms. 2011; 4(4):307-333.
Kieffer, John. 2011. "A Catalog of Self-Affine Hierarchical Entropy Functions." Algorithms 4, no. 4: 307-333.