## Asymptotic Properties Of Data Compression And Suffix Trees (1993)

Venue: | IEEE Trans. Inform. Theory |

Citations: | 42 - 11 self |

### BibTeX

@ARTICLE{Szpankowski93asymptoticproperties,

author = {Wojciech Szpankowski},

title = {Asymptotic Properties Of Data Compression And Suffix Trees},

journal = {IEEE Trans. Inform. Theory},

year = {1993},

volume = {39},

pages = {1647--1659}

}

### Years of Citing Articles

### OpenURL

### Abstract

Recently, Wyner and Ziv have proved that the typical length of a repeated subword found within the first n positions of a stationary ergodic sequence is (1=h) log n in probability where h is the entropy of the alphabet. This finding was used to obtain several insights into certain universal data compression schemes, most notably the Lempel-Ziv data compression algorithm. Wyner and Ziv have also conjectured that their result can be extended to a stronger almost sure convergence. In this paper, we settle this conjecture in the negative in the so called right domain asymptotic, that is, during a dynamic phase of expanding the data base. We prove -- under an additional assumption involving mixing conditions -- that the length of a typical repeated subword oscillates almost surely (a.s.) between (1=h 1 ) log n and (1=h 2 ) log n where 0 ! h 2 ! h h 1 ! 1. We also show that the length of the nth block in the Lempel-Ziv parsing algorithm reveals a similar behavior. We relate our findings to...

### Citations

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2359 |
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On the complexity of finite sequences
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Linear algorithm for data compression via string matching
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Some asymptotic properties of the entropy of stationary ergodic data source with applications to data compression
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Citation Context ... data base to the new position. This idea can be modeled mathematically in two different fashions that are discussed next. A. Static Model -- Left Domain Asymptotic This is the model of Wyner and Ziv =-=[41]-=-. It is assumed that the subsequence to be compressed fX k g 1 k=0 is always the same (by definition fixed at position k = 0), and the data base fX k g \Gamma1 k=\Gamman expands only to the left. Ther... |

60 |
Asymptotic growth of a class of random trees
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Citation Context ...d rigorously the latter result regarding the average size of a suffix tree. Some related topics were discussed by Guibas and Odlyzko in [16]. Our findings were inspired by the seminal paper of Pittel =-=[31]-=- who considered a typical behavior of a trie constructed from independent words (i.e., independent tries). Pittel was the first who noticed that the depth of insertion in an independent trie does not ... |

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Citation Context ...ernal structure (i.e., repeated substrings) of the first n symbols of the pattern P . It turns out that this problem can be efficiently solved by means of a suffix tree (cf. [1], [4], [5], [9], [14], =-=[18]-=-, [28], [40]). In particular, recently Chang and Lawler [11] used suffix trees to design an algorithm that on average needs O((jT j=jP j) log jP j) steps to find all occurrences of the pattern P of le... |

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49 |
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Citation Context ... matching algorithms the longest suffix that matches a substring of the pattern string is used for "fast" shift of the pattern over a text string (cf. Knuth-Morris-Pratt and Boyer-Moore [2];=-= see also [11]-=-), and so forth. The problem of repeated patterns is studied here in a probabilistic framework. We assume that a stationary and ergodic source of information generates an infinite sequence fX k g 1 k=... |

40 |
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Citation Context ...ined for the Markovian model. This is due to two facts: (i) the limiting behavior of independent tries do not differ too much from asymptotics of suffix trees (cf. [18]); (ii) Jacquet and Szpankowski =-=[17]-=- established the limiting distribution of the depth for independent tries in a Markovian framework. (iv) Second Order Behavior for the Lempel-Ziv Parsing Scheme. The limiting distribution of a randoml... |

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Citation Context ...f the height was recently obtained by Devroye, Szpankowski and Rais [14], and the limiting distribution of the depth in a suffix tree is reported in Jacquet and Szpankowski [18]. Arratia and Waterman =-=[6]-=- investigated a related problem, namely the longest contiguous matching within a single sequence, and obtained several interesting results in this direction. Their findings are related to the hight of... |

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A Universal Algorithm for Sequential Data
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Citation Context ...and other kinds of regularities in words. In data compression, such a repeated subsequence can be used to reduce the size of the original sequence (e.g., universal data compression schemes [7], [27], =-=[43]). In exac-=-t string matching algorithms the longest suffix that matches a substring of the pattern string is used for "fast" shift of the pattern over a text string (cf. Knuth-Morris-Pratt and Boyer-Mo... |

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Sample converses in source coding theory
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Citation Context ...scanty in the literature. To our best knowledge, asymptotic analysis of universal data compressions was pursued by Ziv and Lempel (cf. [43], [27]; see also [7]), Wyner and Ziv [41], [42], and Kieffer =-=[22]-=-. The average case analysis of suffix trees was initialized by Grassberger [15], and Apostolico and Szpankowski [5]. For the Bernoulli model, the asymptotic behavior of the height was recently obtaine... |

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Subadditive processes, École d’Été de Probabilités de Saint-Flour, V
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- 1976
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Citation Context ... for example, for the Bernoulli model, the Markovian model, the hidden Markov model and m-dependent model (cf. [13]). In order to apply the Borel-Cantelli Lemma, we use the trick suggested by Kingman =-=[23]-=-; that is, we construct a subsequence n r of n for which O(1= p log n r ) is summable. Fix s, and define n r = 2 s 2 2 2r . Note that e L nr = log n r ! 1=h (a.s.) provided P (B n ) is summable. To pr... |

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Citation Context ...P V i;j=1 �� i p i;j log p i;j where �� i is the stationary distribution of the Markov chain. The other quantities, that is, h 1 and h 2 , are a little harder to evaluate. Pittel [31] and Szpa=-=nkowski [38]-=- evaluated the height of regular tries with Markovian dependency, and they showed that the parameter h 2 is a function of the largest eigenvalue ` of the matrix P [2] = P ffi P which represents the Sc... |

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A note on the height of suffix trees
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- 1992
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Citation Context ...he internal structure (i.e., repeated substrings) of the first n symbols of the pattern P . It turns out that this problem can be efficiently solved by means of a suffix tree (cf. [1], [4], [5], [9], =-=[14]-=-, [18], [28], [40]). In particular, recently Chang and Lawler [11] used suffix trees to design an algorithm that on average needs O((jT j=jP j) log jP j) steps to find all occurrences of the pattern P... |

20 |
Fixed data base version of the Lempel-Ziv data compression algorithm
- WYNER, ZIV
- 1991
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Citation Context ...ssions are rather scanty in the literature. To our best knowledge, asymptotic analysis of universal data compressions was pursued by Ziv and Lempel (cf. [43], [27]; see also [7]), Wyner and Ziv [41], =-=[42]-=-, and Kieffer [22]. The average case analysis of suffix trees was initialized by Grassberger [15], and Apostolico and Szpankowski [5]. For the Bernoulli model, the asymptotic behavior of the height wa... |

15 |
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Citation Context ...ility (pr:) (1:3b) where h is the entropy of X. This result concerns the convergence in probability (pr.) of e L n . In fact, a similar results also holds for L n in the right domain asymptotics (cf. =-=[35]-=-, [39]). Wyner and Ziv [41] asked whether it can be extended to a stronger almost sure (a.s.) convergence. In the right domain asymptotic, we shall settle this question in the negative for the Markovi... |

15 |
Some results on V -ary asymmetric tries
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Citation Context ...); (2:19b) for some " ? 0, where H 2 = P V i=1 p 2 i log p i , and P 1 (x) and P 2 (x) are fluctuating periodic functions with small amplitudes (an explicit formula for the constant C can be foun=-=d in [37]-=-). We conjecture that the same type of limiting distributions can be obtained for the Markovian model. This is due to two facts: (i) the limiting behavior of independent tries do not differ too much f... |

13 |
Odlyzko - String Overlaps
- Guibas, M
- 1981
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Citation Context ...ns an oscillating term). Jacquet and Szpankowski [18] established rigorously the latter result regarding the average size of a suffix tree. Some related topics were discussed by Guibas and Odlyzko in =-=[16]-=-. Our findings were inspired by the seminal paper of Pittel [31] who considered a typical behavior of a trie constructed from independent words (i.e., independent tries). Pittel was the first who noti... |

10 |
A diffusion limit for a class of random-growing binary trees, Prob
- Aldous, Shields
- 1988
(Show Context)
Citation Context ...r independent tries, we obtain the desired result. Finally, we can easily prove our conjecture for a modified version of the Lempel-Ziv parsing algorithm that we already discussed in Example 1.2 (cf. =-=[3]-=-) in which the phrases do not overlap. Such a parsing scheme can be conveniently modeled by another digital tree, namely the so called digital search tree (cf. [3], [1], [26]). Then the length l n of ... |

10 |
Counts of long aligned word matches among random letter sequences
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- 1987
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Citation Context ... P 2 (w k ) 1 A 3=2 = c 1 (EP (w k )) 3=2 ; where (A) follows directly from (3.9) (by setting w k = w 0 k ), and (B) is a consequence of the following inequality, which can be found in Karlin and Ost =-=[20]-=- and Szpankowski [38], `sr ) 0 @ X W k P ` (w k ) 1 A 1=` 0 @ X W k P r (w k ) 1 A 1=r : (3:10) Finally, the above implies the following estimate for the second sum in the denominator of (3.8) X (i;j)... |

9 |
Optimization of stationary control of discrete deterministic processes
- Romanovski
- 1967
(Show Context)
Citation Context ...rix P [2] = P ffi P which represents the Schur product of P (i.e., elementwise product). More precisely, h 2 = (1=2) log ` \Gamma1 . With respect to h 1 we need a result from digraphs (cf. Romanovski =-=[34]-=-, Karp [21]). Consider a digraph on \Sigma with weights equal to \Gamma log p ij where ! i ; ! j 2 \Sigma. Define a cycle C = f! 1 ; ! 2 ; :::; ! v ; ! 1 g for some vsV such that ! i 2 \Sigma, and let... |

7 |
New asymptotic bounds and improvements on the Lempel-Ziv data compression algorithm
- Bender, Wolf
- 1991
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Citation Context ...substrings and other kinds of regularities in words. In data compression, such a repeated subsequence can be used to reduce the size of the original sequence (e.g., universal data compression schemes =-=[7], [27], [4-=-3]). In exact string matching algorithms the longest suffix that matches a substring of the pattern string is used for "fast" shift of the pattern over a text string (cf. Knuth-Morris-Pratt ... |

6 |
The Myriad Virtues of Suffix Trees, Combinatorial Algorithms on Words
- Apostolico
- 1985
(Show Context)
Citation Context ... how to construct such a 1 There is another way of parsing a sequence in which phrases do not overlap. For example, our sequence parsing by using a special data structure called suffix tree (cf. [1], =-=[4]-=-, [15]). Naturally, one is interested in the length of a block in the parsing algorithm. Let l n be the length of the nth block in the Lempel-Ziv parsing algorithm. There is a relationship between l n... |

5 |
On the application of the Borel-Cantelli
- Chung, ErdGs
- 1952
(Show Context)
Citation Context ...in such a way that the probability of the RHS of the above will be easier to evaluate than the original probability. In order to estimate PrfHmskg we use the second moment method (cf. Chung and Erdos =-=[10]), wh-=-ich states that for events A i Prf m [ i=1 A i gs( P m i=1 PrfA i g) 2 P m i=1 PrfA i g + P i6=j PrfA i " A j g : (3:7) In our case, we set A i;j = fC i;jskg, and hence PrfHmskg = Prf S m i;j=1 A... |

4 |
Self-alignments in Words and Their
- Apostolico, Szpankowski
- 1992
(Show Context)
Citation Context ...pends on the internal structure (i.e., repeated substrings) of the first n symbols of the pattern P . It turns out that this problem can be efficiently solved by means of a suffix tree (cf. [1], [4], =-=[5]-=-, [9], [14], [18], [28], [40]). In particular, recently Chang and Lawler [11] used suffix trees to design an algorithm that on average needs O((jT j=jP j) log jP j) steps to find all occurrences of th... |

4 |
On the Variance of the External
- Kirschenhofer, Prodinger, et al.
- 1989
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Citation Context ...1 L n (m). Even in the Bernoulli model major difficulties arise in the evaluation of the limiting distribution of E n in a suffix tree. For independent tries, Kirschenhofer, Prodinger and Szpankowski =-=[24] rec-=-ently obtained for the symmetric alphabet the variance of E n which is varE n = (ff + P 3 (log n))n +O(log 2 n) where ff �� 4:35 : : : (explicit formula for ff can be found in [24]) and P 3 (log n... |

3 |
A Characterization of the Minimum Cycle Mean
- Karp
- 1978
(Show Context)
Citation Context ... P ffi P which represents the Schur product of P (i.e., elementwise product). More precisely, h 2 = (1=2) log ` \Gamma1 . With respect to h 1 we need a result from digraphs (cf. Romanovski [34], Karp =-=[21]-=-). Consider a digraph on \Sigma with weights equal to \Gamma log p ij where ! i ; ! j 2 \Sigma. Define a cycle C = f! 1 ; ! 2 ; :::; ! v ; ! 1 g for some vsV such that ! i 2 \Sigma, and let `(C) = \Ga... |

1 |
On the Lempel-Ziv Parsing Algorithm and Its Digital Tree Representation, INRIA Rapport de Recherche
- Jacquet, Szpankowski
- 1992
(Show Context)
Citation Context ...n for the number of pharses M n and the internal path length of the associated digital tree, however, without explicit formula for the variance. This was recently rectified by Jacquet and Szpankowski =-=[19] who use-=-d a result of Kirschenhofer, Prodinger and Szpankowski [25] to show that var M n �� (fi + P 4 (log n))n= log 3 n where fi �� 0:26600 : : : (explict but complicated formula on fi can be found i... |