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33
Rank modulation for flash memories
, 2009
"... We explore a novel data representation scheme for multilevel flash memory cells, in which a set of n cells stores information in the permutation induced by the different charge levels of the individual cells. The only allowed chargeplacement mechanism is a “pushtothetop” operation, which takes a ..."
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Cited by 44 (24 self)
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We explore a novel data representation scheme for multilevel flash memory cells, in which a set of n cells stores information in the permutation induced by the different charge levels of the individual cells. The only allowed chargeplacement mechanism is a “pushtothetop” operation, which takes a single cell of the set and makes it the topcharged cell. The resulting scheme eliminates the need for discrete cell levels, as well as overshoot errors, when programming cells. We present unrestricted Gray codes spanning all possible ncell states and using only “pushtothetop” operations, and also construct balanced Gray codes. One important application of the Gray codes is the realization of logic multilevel cells, which is useful in conventional storage solutions. We also investigate rewriting schemes for random data modification. We present both an optimal scheme for the worst case rewrite performance and an approximation scheme for the averagecase rewrite performance.
Constructing Trees in Parallel
, 1989
"... O(log = log n processor as well as O(log n) = log n processor CREW deterministic parallel algorithms are presented for constructing Huffman codes from a given list of frequencies. The time can be reduced to O(logn(log log n) ) on a CRCW model, using only n processors. Also presented i ..."
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Cited by 17 (1 self)
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O(log = log n processor as well as O(log n) = log n processor CREW deterministic parallel algorithms are presented for constructing Huffman codes from a given list of frequencies. The time can be reduced to O(logn(log log n) ) on a CRCW model, using only n processors. Also presented is an optimal O(log n) time, O(n= log n) processor EREW parallel algorithm for constructing a tree given a list of leaf depths when the depths are monotonic.
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 the co ..."
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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...
Robust Universal Complete Codes for Transmission and Compression
 Discrete Applied Mathematics
, 1996
"... Several measures are defined and investigated, which allow the comparison of codes as to their robustness against errors. Then new universal and complete sequences of variablelength codewords are proposed, based on representing the integers in a binary Fibonacci numeration system. Each sequence is ..."
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Cited by 10 (4 self)
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Several measures are defined and investigated, which allow the comparison of codes as to their robustness against errors. Then new universal and complete sequences of variablelength codewords are proposed, based on representing the integers in a binary Fibonacci numeration system. Each sequence is constant and need not be generated for every probability distribution. These codes can be used as alternatives to Huffman codes when the optimal compression of the latter is not required, and simplicity, faster processing and robustness are preferred. The codes are compared on several "reallife" examples. 1. Motivation and Introduction Let A = fA 1 ; A 2 ; \Delta \Delta \Delta ; An g be a finite set of elements, called cleartext elements, to be encoded by a static uniquely decipherable (UD) code. For notational ease, we use the term `code' as abbreviation for `set of codewords'; the corresponding encoding and decoding algorithms are always either given or clear from the context. A code i...
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.
A Version Numbering Scheme with a Useful Lexicographical Order
, 1995
"... We describe a numbering scheme for versions with alternatives that has a useful lexicographical ordering. The version hierarchy is a tree. By inspection of the version numbers, we can easily determine whether one version is an ancestor of another. If so, we can determine the version sequence between ..."
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Cited by 6 (2 self)
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We describe a numbering scheme for versions with alternatives that has a useful lexicographical ordering. The version hierarchy is a tree. By inspection of the version numbers, we can easily determine whether one version is an ancestor of another. If so, we can determine the version sequence between these two versions. If not, we can determine the most recent common ancestor to these two versions (i.e., the least upper bound, lub). Sorting the version numbers lexicographically results in a version being followed by all descendants and preceded by all its ancestors. We use a representation of nonnegative integers that is self delimiting and whose lexicographical ordering matches the ordering by value. 1 Introduction Our motivation is a project in collaborative design for Civil Engineering being carried out at Stanford. The approach taken by the project, named CEDB (Collaborative Environment for the Design of Buildings) , leads to an interesting problem in version numbering. Our approa...
Non prefixfree codes for constrained sequences
 in International Symposium on Information Theory, 2005. ISIT 2005, IEEE
"... Abstract — In this paper we consider the use of variable length non prefixfree codes for coding constrained sequences of symbols. We suppose to have a Markov source where some state transitions are impossible, i.e. the stochastic matrix associated with the Markov chain has some null entries. We sho ..."
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Cited by 6 (0 self)
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Abstract — In this paper we consider the use of variable length non prefixfree codes for coding constrained sequences of symbols. We suppose to have a Markov source where some state transitions are impossible, i.e. the stochastic matrix associated with the Markov chain has some null entries. We show that classic Kraft inequality is not a necessary condition, in general, for unique decodability under the above hypothesis and we propose a relaxed necessary inequality condition. This allows, in some cases, the use of non prefixfree codes that can give very good performance, both in terms of compression and computational efficiency. Some considerations are made on the relation between the proposed approach and other existing coding paradigms. I.
Twenty (or so) questions: Dary boundedlength Huffman coding,” preprint available from http://arxiv.org/abs/cs.IT/0602085
"... Abstract — The game of Twenty Questions has long been used to illustrate binary source coding. Recently, a physical device has been developed that mimics the process of playing Twenty Questions, with the device supplying the questions and the user providing the answers. However, this game differs fr ..."
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Cited by 3 (2 self)
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Abstract — The game of Twenty Questions has long been used to illustrate binary source coding. Recently, a physical device has been developed that mimics the process of playing Twenty Questions, with the device supplying the questions and the user providing the answers. However, this game differs from Twenty Questions in two ways: Answers need not be only “yes ” and “no, ” and the device continues to ask questions beyond the traditional twenty; typically, at least 20 and at most 25 questions are asked. The nonbinary variation on source coding is one that is well known and understood, but not with such bounds on length. An upper bound on the related property of fringe, the difference between the lengths of the longest and the shortest codewords, has been considered, but no polynomialtime algorithm currently finds optimal fringelimited codes. An O(n(lmax − lmin))time O(n)space PackageMergebased algorithm is presented here for finding an optimal Dary (binary or nonbinary) source code with all n codeword lengths (numbers of questions) bounded to be within the interval [lmin, lmax]. This algorithm minimizes average codeword length or, more generally, any other quasiarithmetic convex coding penalty. In the case of minimizing average codeword length, time complexity can often be improved via an alternative graphbased reduction. This has, as a special case, a method for nonbinary lengthlimited Huffman coding, which was previously solved via dynamic programming with O(n 2 lmax log D) time and O(n 2 log D) space. These algorithms can also be used to efficiently find a code that is optimal given a limit on fringe. I.
Bounds of redundancy estimates for wordbased encoding of sequences produced by a Bernoulli source
 Russian), Problemy Peredachi Informacii
, 1972
"... The efficiency of a code is estimated by its redundancy R, while the complexity of a code is estimated by its average delay ¯N. In this work we construct wordbased codes, for which R � ¯ N −5/3. Therefore, wordbased codes can attain the same redundancy as blockcodes while being much less complex ..."
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Cited by 3 (0 self)
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The efficiency of a code is estimated by its redundancy R, while the complexity of a code is estimated by its average delay ¯N. In this work we construct wordbased codes, for which R � ¯ N −5/3. Therefore, wordbased codes can attain the same redundancy as blockcodes while being much less complex. We also consider uniform on the output codes, the benefit of which is the lack of a running synchronization error. For such codes ¯ N −1 � R � ¯ N −1, except for a case when all input symbols are equiprobable, when R � ¯ N −2 for infinitely many ¯ N. 1