Conditions for Optimality of the Huffman Algorithm

Parker, D. Stott

doi:10.1137/0209035

Cited by 44 publications

(34 citation statements)

References 19 publications

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“…First, we have provided an expression for the optimal code for the case with exponential penalization (theorem 5) in terms of its classical counterpart [7,24,25]. We have shown that this penalization affects the optimal code in such a way that the Rényi entropy of the source bounds the t-exponential average codeword length (theorem 6).…”

Section: Discussionmentioning

confidence: 99%

“…k tℓ d and where the classical optimal code turns to be {c opt t (i)}. This last one can be computed by the algorithms proposed in [7,24,25].…”

Section: Proof Let Us First Introduce the Density Operatormentioning

confidence: 99%

“…Because ℓ t (C opt t (ρ)) ≤ ℓ t (C Sh t (ρ)) by definition of the optimal code, the upper bound in (24) immediately follows from this inequality.…”

mentioning

confidence: 97%

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Lossless quantum data compression with exponential penalization: an operational interpretation of the quantum Rényi entropy

Bellomo

Bosyk

Holik

et al. 2017

Sci Rep

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Based on the problem of quantum data compression in a lossless way, we present here an operational interpretation for the family of quantum Rényi entropies. In order to do this, we appeal to a very general quantum encoding scheme that satisfies a quantum version of the Kraft-McMillan inequality. Then, in the standard situation, where one is intended to minimize the usual average length of the quantum codewords, we recover the known results, namely that the von Neumann entropy of the source bounds the average length of the optimal codes. Otherwise, we show that by invoking an exponential average length, related to an exponential penalization over large codewords, the quantum Rényi entropies arise as the natural quantities relating the optimal encoding schemes with the source description, playing an analogous role to that of von Neumann entropy.

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Section: Discussionmentioning

confidence: 99%

“…k tℓ d and where the classical optimal code turns to be {c opt t (i)}. This last one can be computed by the algorithms proposed in [7,24,25].…”

Section: Proof Let Us First Introduce the Density Operatormentioning

confidence: 99%

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Lossless quantum data compression with exponential penalization: an operational interpretation of the quantum Rényi entropy

Bellomo

Bosyk

Holik

et al. 2017

Sci Rep

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“…The optimality of this is shown in [12] and can illustrated with an exchange argument (e.g., [13, pp. 124-125] for the linear case).…”

Section: Boundsmentioning

confidence: 99%

On the redundancy of Huffman codes with exponential objectives

Baer¹

2013

2013 IEEE International Symposium on Information Theory

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We present new lower and upper bounds for the compression rate of binary prefix codes over memoryless sources optimized according to two related exponential codeword length objectives. The objectives explored here are exponential-average length and exponential-average redundancy. The first of these relates to various problems involving queueing, uncertainty, and lossless communications. It can be reduced to the second, which has properties more amenable to analysis. These bounds, some of which are tight, are in terms of a form of entropy and/or the probability of an input symbol, improving on recently discovered bounds of similar form. We also observe related properties of optimal codes over the exponential-average redundancy utility.

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“…One such variation of the Huffman algorithm was used in Humblet's dissertation [22] for a queueing application (and further discussed in [18], [19], [23]). The problem this variation solves is as follows: Given probability mass function p and a > 1, find a code minimizing…”

Section: Introductionmentioning

confidence: 99%

Tight bounds on minimum maximum pointwise redundancy

Baer

2008

2008 IEEE International Symposium on Information Theory

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This paper presents new lower and upper bounds for the optimal compression of binary prefix codes in terms of the most probable input symbol, where compression efficiency is determined by the nonlinear codeword length objective of minimizing maximum pointwise redundancy. This objective relates to both universal modeling and Shannon coding, and these bounds are tight throughout the interval. The upper bounds also apply to a related objective, that of d th exponential redundancy.

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Conditions for Optimality of the Huffman Algorithm

Cited by 44 publications

References 19 publications

Lossless quantum data compression with exponential penalization: an operational interpretation of the quantum Rényi entropy

Lossless quantum data compression with exponential penalization: an operational interpretation of the quantum Rényi entropy

On the redundancy of Huffman codes with exponential objectives

Tight bounds on minimum maximum pointwise redundancy

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