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A universal algorithm for sequential data compression
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 1977
"... A universal algorithm for sequential data compression is presented. Its performance is investigated with respect to a nonprobabilistic model of constrained sources. The compression ratio achieved by the proposed universal code uniformly approaches the lower bounds on the compression ratios attainabl ..."
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Cited by 1501 (7 self)
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A universal algorithm for sequential data compression is presented. Its performance is investigated with respect to a nonprobabilistic model of constrained sources. The compression ratio achieved by the proposed universal code uniformly approaches the lower bounds on the compression ratios attainable by blocktovariable codes and variabletoblock codes designed to match a completely specified source.
Huffman coding with unequal letter costs (Extended Abstract)
 IN: PROCEEDINGS OF THE THIRYFOURTH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING, ACM
, 2002
"... In the standard Huffman coding problem, one is given a set of words and for each word a positive frequency. The goal is to encode each word w as a codeword c(w) over a given alphabet. The encoding must be prefixfree (no codeword is a prefixof any other) and should minimize the weighted average codew ..."
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Cited by 17 (5 self)
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In the standard Huffman coding problem, one is given a set of words and for each word a positive frequency. The goal is to encode each word w as a codeword c(w) over a given alphabet. The encoding must be prefixfree (no codeword is a prefixof any other) and should minimize the weighted average codeword size � w freq(w) c(w). The problem has a wellknown polynomialtime algorithm due to Huffman [15]. Here we consider the generalization in which the letters of the encoding alphabet may have nonuniform lengths. The goal is to minimize the weighted average codeword length w freq(w) cost(c(w)), where cost(s) is the sum of the (possibly nonuniform) lengths of the letters in s. Despitemuch previous work, the problem is not known to be NPhard, nor was it previously known to have a polynomialtime approximation algorithm. Here we describe a polynomialtime approximation scheme (PTAS) for the problem.
Optimal PrefixFree Codes for Unequal Letter Costs: Dynamic Programming with the Monge Property
 J. Algorithms
, 2000
"... In this paper we discuss the problem of finding optimal prefixfree codes for unequal letter costs, a variation of the classical Huffman coding problem. Our problem consists of finding a minimal cost prefixfree code in which the encoding alphabet consists of unequal cost (length) letters, with leng ..."
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Cited by 17 (7 self)
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In this paper we discuss the problem of finding optimal prefixfree codes for unequal letter costs, a variation of the classical Huffman coding problem. Our problem consists of finding a minimal cost prefixfree code in which the encoding alphabet consists of unequal cost (length) letters, with lengths ff and fi. The most efficient algorithm known previously requires O(n 2+max(ff;fi) ) time to construct such a minimalcost set of n codewords, provided ff and fi are integers. In this paper we provide an O(n max(ff;fi) ) time algorithm. Our improvement comes from the use of a more sophisticated modeling of the problem, combined with the observation that the problem possesses a "Monge property" and that the SMAWK algorithm on monotone matrices can therefore be applied. Keywords: Dynamic Programming, Huffman Codes, Lopsided Trees, Monge Matrix, Monotone Matrix, PrefixFree Codes. 1 Introduction Finding optimal prefixfree codes for unequal letter costs (and the associated problem of...
Matching dyadic distributions to channels
 in Proc. Data Compression Conf., 2011
"... Many communication channels with discrete input have nonuniform capacity achieving probability mass functions (PMF). By parsing a stream of independent and equiprobable bits according to a full prefixfree code, a modulator can generate dyadic PMFs at the channel input. In this work, we show that ..."
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Cited by 11 (5 self)
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Many communication channels with discrete input have nonuniform capacity achieving probability mass functions (PMF). By parsing a stream of independent and equiprobable bits according to a full prefixfree code, a modulator can generate dyadic PMFs at the channel input. In this work, we show that for discrete memoryless channels and for memoryless discrete noiseless channels, searching for good dyadic input PMFs is equivalent to minimizing the KullbackLeibler distance between a dyadic PMF and a weighted version of the capacity achieving PMF. We define a new algorithm called Geometric Huffman Coding (GHC) and prove that GHC finds the optimal dyadic PMF in O(m logm) steps where m is the number of input symbols of the considered channel. Furthermore, we prove that by generating dyadic PMFs of blocks of consecutive input symbols, GHC achieves capacity when the block length goes to infinity. I.
Optimal Parsing Trees for RunLength Coding of Biased Data
 IEEE Int. Symposium on Information Theory
, 2006
"... Abstract — We study coding schemes which encode unconstrained sequences into runlengthlimited (d, k)constrained sequences. We present a general framework for the construction of such (d, k)codes from variablelength source codes. This framework is an extension of the previously suggested bit stu ..."
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Abstract — We study coding schemes which encode unconstrained sequences into runlengthlimited (d, k)constrained sequences. We present a general framework for the construction of such (d, k)codes from variablelength source codes. This framework is an extension of the previously suggested bit stuffing, bit flipping and symbol sliding algorithms. We show that it gives rise to new code constructions which achieve improved performance over the three aforementioned algorithms. Therefore, we are interested in finding optimal codes under this framework, optimal in the sense of maximal achievable asymptotic rates. However, this appears to be a difficult problem. In an attempt to solve it, we are led to consider the encoding of unconstrained sequences of independent but biased (as opposed to equiprobable) bits. Here, our main result is that one can use the Tunstall source coding algorithm to generate optimal codes for a partial class of (d, k) constraints.
Lopsided Trees: Analyses, Algorithms, and Applications
 in Automata, Languages and Programming, Proceedings of the 23rd International Colloquium on Automata, Languages, and Programming (ICALP
, 2000
"... Lopsided trees are rooted, ordered, trees in which the length of an edge from a node to its i th child depends upon the value of i: These trees model a variety of problems and have therefore been extensively studied. In this paper we combine analytic and combinatorial techniques to address three o ..."
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Cited by 3 (1 self)
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Lopsided trees are rooted, ordered, trees in which the length of an edge from a node to its i th child depends upon the value of i: These trees model a variety of problems and have therefore been extensively studied. In this paper we combine analytic and combinatorial techniques to address three open problems on such trees: ffl Given n, efficiently construct a lopsided tree with n leaves that has minimal externalpathlength. ffl Express the cost of the minimal externalpathlength tree as a function of n: ffl Calculate exactly how many nodes of depth x exist in the infinite lopsided tree. Lopsided trees model Varn codes, prefix free codes in which the letters of the encoding alphabet can have different lengths. The solutions to the first and second problems above solve corresponding open problems on Varn codes. The solution to the third problem can be used to model the performance of broadcasting algorithms in the postal model of communication. Finding these solutions requires g...
HUFFMAN CODING WITH LETTER COSTS: A LINEARTIME APPROXIMATION SCHEME
, 2012
"... We give a polynomialtime approximation scheme for the generalization of Huffman coding in which codeword letters have nonuniform costs (as in Morse code, where the dash is twice as long as the dot). The algorithm computes a (1 + ɛ)approximate solution in time O(n + f(ɛ)log 3 n), where n is the in ..."
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Cited by 2 (1 self)
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We give a polynomialtime approximation scheme for the generalization of Huffman coding in which codeword letters have nonuniform costs (as in Morse code, where the dash is twice as long as the dot). The algorithm computes a (1 + ɛ)approximate solution in time O(n + f(ɛ)log 3 n), where n is the input size.
CapacityAchieving Probabilistic Shaping for Noisy and Noiseless Channels
"... Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfügbar. Acknowledgments I want to thank Prof. Rudolf Mathar for the freedom to pursue my ideas during my time at his institute. The TI group was my second home for four years and a half, thank you all. Thanks to Daniel ..."
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Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online verfügbar. Acknowledgments I want to thank Prof. Rudolf Mathar for the freedom to pursue my ideas during my time at his institute. The TI group was my second home for four years and a half, thank you all. Thanks to Daniel and Gernot, Chunhui and Milan, Fabian and Steven, Andreas and Martijn for collaboration, trips around the world, coffee, and friendship. Special thanks to Prof. Valdemar Cardoso da Rocha Junior and Prof. Cecilio Pimentel for the continuous support. I am grateful to my father Prof. Siegfried Böcherer for all the telephone calls that helped me to get the math at least partially right. Prof. Gerhard Kramer read my dissertation cover to cover, which is the best reward I can think of. Finally, I thank my wife Noêmia and our children Izabel and Rafael for reminding me on a daily basis that work is not the only thing that matters. 2
DOI: 10.1007/s0045300100631 Lopsided Trees, I: Analyses 1
, 2001
"... Abstract. Lopsided trees are rooted, ordered trees in which the length of an edge from a node to its ith child depends upon the value of i. These trees model a variety of problems and have therefore been extensively studied. In this paper we combine analytic and combinatorial techniques to address t ..."
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Abstract. Lopsided trees are rooted, ordered trees in which the length of an edge from a node to its ith child depends upon the value of i. These trees model a variety of problems and have therefore been extensively studied. In this paper we combine analytic and combinatorial techniques to address three open problems on such trees: • Given n, characterize the combinatorial structure of a lopsided tree with n leaves that has minimal external path length. • Express the cost of the minimal external path length tree as a function of n. • Calculate exactly how many nodes of depth ≤ x exist in the infinite lopsided tree. Lopsided trees model Varn codes, prefix free codes in which the letters of the encoding alphabet can have different lengths. The solutions to the first and second problems above solve corresponding open problems on Varn codes. The solution to the third problem can be used to model the performance of broadcasting algorithms in the postal model of communication. Finding these solutions requires generalizing the definition of Fibonacci numbers and then using Mellintransform techniques.