Asymptotic behavior of the Lempel-Ziv parsing scheme and digital search trees

Jacquet, Philippe; Szpankowski, Wojciech

doi:10.1016/0304-3975(94)00298-w

Cited by 91 publications

(105 citation statements)

References 23 publications

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“…Let us compute the probability P n = P L(B 2 ) > n 2 − y n + ǫy n for large n. We shall prove that P n → 0. We have P n = P (B n,0 < (1 − ǫ)y n ) (by (3.7)) ≤ P Y n − n 2 < (1 − ǫ)y n (by Lemma 3.5) = P Y n − y n √ Var Y n < −ǫy n + n/2 √ Var Y n < Aµ k (by Theorem 1A of [13])…”

Section: Thusmentioning

confidence: 94%

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Fast Algorithm for Optimal Compression of Graphs

Choi

2010

2010 Proceedings of the Seventh Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

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We consider the problem of finding optimal description for general unlabeled graphs. Given a probability distribution on labeled graphs, we introduced in [4] a structural entropy as a lower bound for the lossless compression of such graphs. Specifically, we proved that the structural entropy for the Erdős-Rényi random graph, in which edges are added with probability p, is`n 2´h (p) − n log n + O(n), where n is the number of vertices and h(p) = −p log p − (1 − p) log(1−p) is the entropy rate of a conventional memoryless binary source. In this paper, we prove the asymptotic equipartition property for such graphs. Then, we propose a faster compression algorithm that asymptotically achieves the structural entropy up to the first two leading terms with high probability. Our algorithm runs in O(n + e) time on average where e is the number of edges. To prove its asymptotic optimality, we introduce binary trees that one can classify as in-between tries and digital search trees. We use analytic techniques such as generating functions, Mellin transform, and poissonization to establish our findings. Our experimental results confirm theoretical results and show the usefulness of our algorithm for real-world graphs such as the Internet, biological networks, and social networks. IntroductionBrooks argues in [3] that there is "no theory that gives us a metric for information embodied in structure." Shannon himself alluded to it fifty years earlier in his little known 1953 paper [20]. In fact, Brooks emphasizes the importance of the quantification of information in physical structure. In computer science, however, it is more important to understand the information embodied in abstract structures that are of our particular interests in this paper. For instance, how can we quantify the amount of information in the structure of graphs such as the Internet, social networks, and biological networks? How can we understand and utilize the "structure" of non-conventional data structures such as biological data, topographical maps, medical data, and volumetric data? As the first step to understanding information in such structures, we focus on structure in graphs.

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Section: Thusmentioning

confidence: 94%

“…It is easy to see that y n represents the expected path length in a digital search tree over n strings as discussed in [13,22]. The authors of [13] proved, among others, that…”

Section: Thusmentioning

confidence: 99%

Fast Algorithm for Optimal Compression of Graphs

Choi

2010

2010 Proceedings of the Seventh Workshop on Analytic Algorithmics and Combinatorics (ANALCO)

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“…Ziv [26] made a conjecture concerning the number of codewords. However, Aldous and Shields [4] were only able to resolve the problem for independent, identically equidistributed processes, and it took careful analysis by Jacquet and Szpankowski [11] to extend their results to independent, identically distributed asymmetric processes. For example, Theorem 1A of [11] is as follows.…”

Section: Definition 13 (Grassberger Prefix) Given An N and Anmentioning

confidence: 99%

A Central Limit Theorem for Non-Overlapping Return Times

Johnson

2006

J. Appl. Probab.

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Define the non-overlapping return time of a block of a random process to be the number of blocks that pass by before the block in question reappears. We prove a central limit theorem based on these return times. This result has applications to entropy estimation, and to the problem of determining if digits have come from an independent, equidistributed sequence. In the case of an equidistributed sequence, we use an argument based on negative association to prove convergence under conditions weaker than those required in the general case.

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“…For example Theorem 1A of [9] shows that Theorem 1.9 Given an binary asymmetric ( P(Z 1 = 0) = 1/2) IID process, then L m , the total length of the m words in a Lempel-Ziv tree, satisfies…”

Section: Other Similar Definitionsmentioning

confidence: 99%

A Central Limit Theorem for Non-Overlapping Return Times

Johnson

2006

Journal of Applied Probability

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Define the non-overlapping return time of a random process to be the number of blocks that we wait before a particular block reappears. We prove a Central Limit Theorem based on these return times. This result has applications to entropy estimation, and to the problem of determining if digits have come from an independent equidistributed sequence. In the case of an equidistributed sequence, we use an argument based on negative association to prove convergence under weaker conditions.

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Asymptotic behavior of the Lempel-Ziv parsing scheme and digital search trees

Cited by 91 publications

References 23 publications

Fast Algorithm for Optimal Compression of Graphs

Fast Algorithm for Optimal Compression of Graphs

A Central Limit Theorem for Non-Overlapping Return Times

A Central Limit Theorem for Non-Overlapping Return Times

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