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53
Is Huffman Coding Dead?
 COMPUTING
, 1993
"... In recent publications about data compression, arithmetic codes are often suggested as the state of the art, rather than the more popular Huffman codes. While it is true that Huffman codes are not optimal in all situations, we show that the advantage of arithmetic codes in compression performance is ..."
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Cited by 17 (3 self)
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In recent publications about data compression, arithmetic codes are often suggested as the state of the art, rather than the more popular Huffman codes. While it is true that Huffman codes are not optimal in all situations, we show that the advantage of arithmetic codes in compression performance is often negligible. Referring also to other criteria, we conclude that for many applications, Huffman codes should still remain a competitive choice.
Bounding the Depth of Search Trees
 The Computer Journal
, 1993
"... For an ordered sequence of n weights, Huffman's algorithm constructs in time and space O(n) a search tree with minimum average path length, or, which is equivalent, a minimum redundancy code. However, if an upper bound B is imposed on the length of the codewords, the best known algorithms for t ..."
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Cited by 16 (5 self)
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For an ordered sequence of n weights, Huffman's algorithm constructs in time and space O(n) a search tree with minimum average path length, or, which is equivalent, a minimum redundancy code. However, if an upper bound B is imposed on the length of the codewords, the best known algorithms for the construction of an optimal code have time and space complexities O(Bn 2 ). A new algorithm is presented, which yields suboptimal codes, but in time O(n log n) and space O(n). Under certain conditions, these codes are shown to be close to optimal, and extensive experiments suggest that in many practical applications, the deviation from the optimum is negligible. 1. Motivation and Introduction We consider the set B(n; b) of extended binary trees with n leaves, labelled 1 to n, and with depth b, henceforth called brestricted trees. An extended binary tree is a binary tree in which every internal node has two sons (here, and in what follows, we use the terminology of Knuth [16, pp. 399...
A fast and spaceeconomical algorithm for lengthlimited coding
 Proc. Int. Symp. Algorithms and Computation, pp.1221
, 1995
"... Abstract. The minimumredundancy prefix code problem is to determine a list of integer codeword lengths I = [li l i E {1... n}], given a list of n symbol weightsp = [pili C {1.n}], such that ~' ~ 2l ' < 1, 9 " i = ln and ~i=1 lipi is minimised. An extension is the minimumre ..."
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Cited by 15 (1 self)
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Abstract. The minimumredundancy prefix code problem is to determine a list of integer codeword lengths I = [li l i E {1... n}], given a list of n symbol weightsp = [pili C {1.n}], such that ~' ~ 2l ' < 1, 9 &quot; i = ln and ~i=1 lipi is minimised. An extension is the minimumredundancy lengthlimited prefix code problem, in which the further constraint li < L is imposed, for all i C {1...n} and some integer L> [log 2 hi. The packagemerge algorithm of Larmore and Hirschberg generates lengthlimited codes in O(nL) time using O(n) words of auxiliary space. Here we show how the size of the work space can be reduced to O(L2). This represents a useful improvement, since for practical purposes L is O(log n). 1
Improved Bounds on the Inefficiency of LengthRestricted Prefix Codes
 Departamento de Inform'atica, PUCRJ, Rio de
, 1997
"... : Consider an alphabet \Sigma = fa 1 ; : : : ; ang with corresponding symbol probabilities p 1 ; : : : ; pn . The L\Gammarestricted prefix code is a prefix code where all the code lengths are not greater than L. The value L is a given integer such that L dlog ne. Define the average code length dif ..."
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Cited by 14 (5 self)
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: Consider an alphabet \Sigma = fa 1 ; : : : ; ang with corresponding symbol probabilities p 1 ; : : : ; pn . The L\Gammarestricted prefix code is a prefix code where all the code lengths are not greater than L. The value L is a given integer such that L dlog ne. Define the average code length difference by ffl = P n i=1 p i :l i \Gamma P n i=1 p i :l i , where l 1 ; : : : ; l n are the code lengths of the optimal Lrestricted prefix code for \Sigma and l 1 ; : : : ; l n are the code lengths of the optimal prefix code for \Sigma. Let / be the golden ratio 1,618. In this paper, we show that ffl ! 1=/ L\Gammadlog(n+dlog ne\GammaL)e\Gamma1 when L ? dlog ne. We also prove the sharp bound ffl ! dlog ne \Gamma 1, when L = dlog ne. By showing the lower bound 1 / L\Gammadlog ne+2+dlog n n\GammaL e \Gamma1 on the maximum value of ffl, we guarantee that our bound is asymptotically tight in the range dlog ne ! L n=2. Furthermore, we present an O(n) time and space 1=/ L\Gammadlo...
The WARMUP Algorithm: A Lagrangean Construction of Length Restricted Huffman Codes
 Departamento de Inform'atica, PUCRJ, Rio de
, 1996
"... : Given an alphabet fa 1 ; : : : ; ang with corresponding set of weights fw 1 ; : : : ; wng, and a number L dlog ne, we introduce an O(n log n+n log w) algorithm for constructing a suboptimal prefix code with restricted maximal length L, where w is the highest presented weight. The number of additi ..."
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Cited by 13 (8 self)
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: Given an alphabet fa 1 ; : : : ; ang with corresponding set of weights fw 1 ; : : : ; wng, and a number L dlog ne, we introduce an O(n log n+n log w) algorithm for constructing a suboptimal prefix code with restricted maximal length L, where w is the highest presented weight. The number of additional bits per symbol generated by our code is not greater than 1=/ L\Gammadlog(n+dlog ne\GammaL)e\Gamma2 when L ? dlog ne + 1, where / is the golden ratio 1:618. An important feature of the proposed algorithm is its implementation simplicity. The algorithm is basically a selected sequence of Huffman trees construction for modified weights. Keywords: Prefix codes, Huffman Trees, Lagragean Duality Resumo: Dado um alfabeto fa 1 ; : : : ; ang com pesos correspondentes fw 1 ; : : : ; wng e um n'umero L dlog ne, n'os apresentamoso um algoritmo de de complexidade O(n log n + n log w)para construit c'odigos de prefixo sub'otimos com restric~ao de comprimento L, onde w 'e o maior peso do dado co...
A general framework for codes involving redundancy minimization
 IEEE Transactions on Information Theory
, 2006
"... Abstract — A framework with two scalar parameters is introduced for various problems of finding a prefix code minimizing a coding penalty function. The framework involves a twoparameter class encompassing problems previously proposed by Huffman [1], Campbell [2], Nath [3], and Drmota and Szpankowsk ..."
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Cited by 9 (6 self)
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Abstract — A framework with two scalar parameters is introduced for various problems of finding a prefix code minimizing a coding penalty function. The framework involves a twoparameter class encompassing problems previously proposed by Huffman [1], Campbell [2], Nath [3], and Drmota and Szpankowski [4]. It sheds light on the relationships among these problems. In particular, Nath’s problem can be seen as bridging that of Huffman with that of Drmota and Szpankowski. This leads to a lineartime algorithm for the last of these with a solution that solves a range of Nath subproblems. We find simple bounds and lineartime Huffmanlike optimization algorithms for all nontrivial problems within the class.
Lossless Compression for Text and Images
 International Journal of High Speed Electronics and Systems
, 1995
"... Most data that is inherently discrete needs to be compressed in such a way that it can be recovered exactly, without any loss. Examples include text of all kinds, experimental results, and statistical databases. Other forms of data may need to be stored exactly, such as imagesparticularly bilevel ..."
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Cited by 7 (0 self)
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Most data that is inherently discrete needs to be compressed in such a way that it can be recovered exactly, without any loss. Examples include text of all kinds, experimental results, and statistical databases. Other forms of data may need to be stored exactly, such as imagesparticularly bilevel ones, or ones arising in medical and remotesensing applications, or ones that may be required to be certified true for legal reasons. Moreover, during the process of lossy compression, many occasions for lossless compression of coefficients or other information arise. This paper surveys techniques for lossless compression. The process of compression can be broken down into modeling and coding. We provide an extensive discussion of coding techniques, and then introduce methods of modeling that are appropriate for text and images. Standard methods used in popular utilities (in the case of text) and international standards (in the case of images) are described. Keywords Text compression, ima...
Efficient Implementation of the WARMUP Algorithm for the Construction of LengthRestricted Prefix Codes
 in Proceedings of the ALENEX
, 1999
"... . Given an alphabet \Sigma = fa1 ; : : : ; ang with a corresponding list of positive weights fw1 ; : : : ; wng and a length restriction L, the lengthrestricted prefix code problem is to find, a prefix code that minimizes P n i=1 w i l i , where l i , the length of the codeword assigned to a i , ..."
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Cited by 5 (0 self)
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. Given an alphabet \Sigma = fa1 ; : : : ; ang with a corresponding list of positive weights fw1 ; : : : ; wng and a length restriction L, the lengthrestricted prefix code problem is to find, a prefix code that minimizes P n i=1 w i l i , where l i , the length of the codeword assigned to a i , cannot be greater than L, for i = 1; : : : ; n. In this paper, we present an efficient implementation of the WARMUP algorithm, an approximative method for this problem. The worstcase time complexity of WARMUP is O(n log n +n log wn ), where wn is the greatest weight. However, some experiments with a previous implementation of WARMUP show that it runs in linear time for several practical cases, if the input weights are already sorted. In addition, it often produces optimal codes. The proposed implementation combines two new enhancements to reduce the space usage of WARMUP and to improve its execution time. As a result, it is about ten times faster than the previous implementat...