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.
Highlights
A traditional type class consists of all permutations of a fixed finite-length data string
Let {Sj : j ≥ 1} be a sequence of hierarchical type classes from Sm,k such that Sj is of order j (j ≥ 1)
A hierarchical source is defined to be a family {S(λ) : λ ∈ Λ(m, k)} in which each S(λ) is a hierarchical type class selected from Sm,k (λ). (We will impose a natural consistency condition on how these selections are made in our formal hierarchical source definition to be given .)
Summary
A traditional type class consists of all permutations of a fixed finite-length data string. Given a hierarchical type class S, there is a simple lossless coding algorithm which encodes each string in S into a fixed-length binary codeword of minimal length, and decodes the string from its codeword This algorithm is simple for the case when the partitioning parameter is k = 2, and we illustrate this case in Example 1 which follows; the case of general k ≥ 2 is discussed in [3]. Each string in S is encoded by visiting, in depth-first order, the non-leaf vertices of its tree representation whose children have different labels. 2: For each j ≥ 0, the distinct hierarchical type classes of order j form a partition of the set of all j-strings.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
