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On the Construction of (Explicit) Khodak’s Code and Its Analysis
 IEEE Trans. Information Theory
"... Variabletovariable codes are very attractive yet not well understood data compression schemes. In 1972 Khodak claimed to provide upper and lower bounds for the achievable redundancy rate, however, he did not offer explicit construction of such codes. In this paper, we first present a constructive ..."
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Variabletovariable codes are very attractive yet not well understood data compression schemes. In 1972 Khodak claimed to provide upper and lower bounds for the achievable redundancy rate, however, he did not offer explicit construction of such codes. In this paper, we first present a constructive and transparent proof of Khodak’s result showing that for memoryless sources there exists a code with the average redundancy rate bounded by D −5/3,whereD is the average delay (e.g., the average length of a phrase). We also describe an algorithm that constructs a variabletovariable length code with a small redundancy rate for large D. Then, we discuss several generalizations. We extend the above result to Markov sources and prove that the worst case redundancy (i.e., for individual sequences) does not exceed D −4/3. Finally, we consider bounds that are valid for almost all memoryless and Markov sources, that is, for almost all sources there exists a variabletovariable code such that its average redundancy rate is bounded by D −4/3−m/3+ε while its worst case redundancy rate by D −1−m/3+ε,wherem is the cardinality of the alphabet. We complete our analysis with a lower bound showing that
Average Redundancy for Known Sources: Ubiquitous Trees in Source Coding
, 2008
"... Analytic information theory aims at studying problems of information theory using analytic techniques of computer science and combinatorics. Following Hadamard’s precept, these problems are tackled by complex analysis methods such as generating functions, Mellin transform, Fourier series, saddle poi ..."
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Analytic information theory aims at studying problems of information theory using analytic techniques of computer science and combinatorics. Following Hadamard’s precept, these problems are tackled by complex analysis methods such as generating functions, Mellin transform, Fourier series, saddle point method, analytic poissonization and depoissonization, and singularity analysis. This approach lies at the crossroad of computer science and information theory. In this survey we concentrate on one facet of information theory (i.e., source coding better known as data compression), namely the redundancy rate problem. The redundancy rate problem determines by how much the actual code length exceeds the optimal code length. We further restrict our interest to the average redundancy for known sources, that is, when statistics of information sources are known. We present precise analyses of three types of lossless data compression schemes, namely fixedtovariable (FV) length codes, variabletofixed (VF) length codes, and variabletovariable (VV) length codes. In particular, we investigate average redundancy of Huffman, Tunstall, and Khodak codes. These codes have succinct representations as trees, either as coding or parsing trees, and we analyze here some of their parameters (e.g., the average path from the root to a leaf).
Efficient Implementation of the Generalized Tunstall Code Generation Algorithm
, 809
"... Abstract — A method is presented for constructing a Tunstall code that is linear time in the number of output items. This is an improvement on the state of the art for nonBernoulli sources, including Markov sources, which require a (suboptimal) generalization of Tunstall’s algorithm proposed by Sav ..."
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Abstract — A method is presented for constructing a Tunstall code that is linear time in the number of output items. This is an improvement on the state of the art for nonBernoulli sources, including Markov sources, which require a (suboptimal) generalization of Tunstall’s algorithm proposed by Savari and analytically examined by Tabus and Rissanen. In general, if n is the total number of output leaves across all Tunstall trees, s is the number of trees (states), and D is the number of leaves of each internal node, then this method takes O((1 + (log s)/D)n) time and O(n) space. I.