Variable-length-to-variable length source coding: a greedy step-by-step algorithm

Fabris, Francesco

doi:10.1109/18.149517

Cited by 17 publications

(15 citation statements)

References 4 publications

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“…We showed that the proposed AIFV codes can attain better compression than Huffman codes for three kinds of sources, P 0 , P 1 , and P 2 . We note that Fabris [4] considered a variable-to-variable-length code (VV code) constructed by concatenating a Huffman code to a Tunstall code. It might be possible to construct a good VV code by concatenating a binary relaxed AIFV code to an AIVF code.…”

Section: Discussionmentioning

confidence: 99%

Almost instantaneous FV codes

Yamamoto

Wei

2013

2013 IEEE International Symposium on Information Theory

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Section: Discussionmentioning

confidence: 99%

Almost instantaneous FV codes

Yamamoto

Wei

2013

2013 IEEE International Symposium on Information Theory

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“…The most interesting, as already mentioned, is a thirty year old work by Khodak [14]. To the best of our knowledge not much was done since then, except that Fabris [9] (cf. also [10,18]) analyzed Tunstall-Huffman VV code and provided a simple bound on their redundancy rate.…”

Section: Introductionmentioning

confidence: 99%

“…While it is well known that every VV (prefix) code is a concatenation of a variable-to-fixed length code (e.g., Tunstall code) and a fixed-to-variable length encoding (e.g., Huffman code), an optimal VV code has not yet been found. Fabris [9] proved that greedy, step by step, optimization (that is, a concatenation of Tunstall and Huffman codes) does not lead to an optimal VV code. In order to assess performance of VV codes, one needs to evaluate (at least asymptotically) the redundancy rate of (optimal) VV codes, which is still unknown.…”

Section: Introductionmentioning

confidence: 99%

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On the Construction of (Explicit) Khodak's Code and Its Analysis

Bugeaud

Drmota

Szpankowski

2008

IEEE Trans. Inform. Theory

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Variable-to-variable 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 bounded by D −5/3 , where D is the average delay (e.g., the average length of a dictionary entry). We also describe an algorithm that constructs a variable-to-variable length code with a small redundancy rate for large D. Then, we discuss several generalizations. We prove that the worst case redundancy does not exceed D −4/3 . Furthermore we provide similar upper bounds for Markov sources (of order 1). Finally, we consider bounds that are valid for almost all memoryless and Markov sources for which the set of exceptional source parameters has zero measure. In particular, for all memoryless sources outside this exceptional class, we prove there exists a variable-to-variable code with the average redundancy rate bounded by D −4/3−m/3+ε and the worst case redundancy rate bounded by D −1−m/3+ε , where m is the cardinality of the alphabet. We complete our analysis with a lower bound showing that for all variable-to-variable codes the average and the worst case redundancy rates are at least D −2m−1−ε for almost all memoryless sources in the sense that the set of exceptional source parameters has zero measure. We prove these results using techniques of Diophantine approximations.Index Terms: Variable-to-variable length codes, average and maximal redundancy rates, metric Diophantine approximations. * A preliminary version of some parts of this paper was presented at

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“…Also, Stubley [7] has demonstrated that the optimal dual-tree codes, obtained by exhaustive search, are only slightly better than the heuristically designed ones described below. In a recent, similar analysis of variable-to-variable-length codes, Fabris [2] has also considered greedy extensions to minimize the resulting rate of the Huffman code.…”

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confidence: 98%

Divergence and the construction of variable-to-variable-length lossless codes by source-word extensions

Freeman

[Proceedings] DCC `93: Data Compression Conference

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Variable-to-variable-length lossless codes a r e described using dual leaf-linked trees: one specifying t h e parsing of t h e source symbols into source words and t h e other specifying t h e formation of code words from code symbols. Compression exceeds entropy by t h e amount of t h e informational divergence, between source words and code words, divided by t h e expected source-word length. The asymptotic optimality of Tunstall or Huffman codes derives from t h e bounding of divergence while t h e expected source-word length is made arbitrarily large. A heuristic extension scheme is asymptotically optimal but also acts to reduce t h e divergence by retaining those source words which a r e well matched t o their corresponding code words. DUAL-TREE ENTROPY CODING [3][4]Variable-length source words are specified by a complete, proper, labelled, a-ary parse tree. Its symbol alphabet SW, with lSwl = a, is that of the source sequence w. Its index alphabet is Iw = (0, ,7' -1 ) . To parse one source word at the encoder, symbols of the source sequence specify a path from the root node of the parse tree until a leaf node is reached. To reconstruct one source word at the decoder, a leaf node is specified and the source symbols are recreated by following its unique path from the parse tree root.Variable-length code words are similarly specified by a complete, proper, labelled, p-ary code tree. Its symbol alphabet Sz, with lSzl = p, is that of the channel sequence z. Its index alphabet is I z = (0, --,7). Code words are parsed from the channel sequence by the decoder or constructed by the encoder using root-to-leaf paths in the code tree.A one-to-one interface is established between the 7' words of the parse tree and a subset of the 7 words of the code tree. To avoid unused code words, it is assumed that the parse tree has 7 -7' fake leaf nodes representing arbitrary source words of tThis work

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Variable-length-to-variable length source coding: a greedy step-by-step algorithm

Cited by 17 publications

References 4 publications

Almost instantaneous FV codes

Almost instantaneous FV codes

On the Construction of (Explicit) Khodak's Code and Its Analysis

Divergence and the construction of variable-to-variable-length lossless codes by source-word extensions

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