A. Lempel and J. Ziv, On the complexity of finite sequences. IEEE Trans. Inform. Theory, vol. IT-22, no. 1, pp. 75--81, January 1976.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that one lacks a plausible monotonicity property. 1 Introduction In this paper we consider two measures of complexity defined on finite sequences over finite alphabets. The measures were suggested by a definition in the seminal paper of Lempel and Ziv, On the Complexity of Finite Sequences. [LempelZ 76] The definition, in the proof of Theorem 2, reads Let N denote the maximum possible number of distinct words into which a sequence of length n over A can be parsed. Reading with Talmudic scruple, one can understand N to count either 1) the number of phrases in a parsing into distinct phrases, or ....

....again with three distinct phrases among a total of four. Either interpretation has the appealing property of being invariant under sequence reversal, a property most appropriate to the study of finite combinatorial objects rather than of sources. The complexities defined by extremal parsing [LempelZ 76]: Parse the longest string which can be found also to begin at an earlier point; or by incremental parsing [ZivL 78] Parse the shortest string which has not yet been parsed are not invariant under sequence reversal. Indeed Gilbert and Kadota believe that under the latter measure, the ....

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A. Lempel and J. Ziv "On the Complexity of Finite Sequences", IEEE Transactions on Information Theory 22:1, 75-81. (1976)

....or recency rank encoding for messages selected independently from A with those probabilities, but will do worse when selections of each message type cluster in time. Vitter [20] gives a modification with better worst case results on arbitrary sequences of input symbols. The Ziv and Lempel schemes [13], 23] 24] 25] are much more sophisticated and powerful, encoding message sequences of increasing length and coping with redundancies of higher order. Modified and augmented versions of the Ziv and Lempel schemes have been implemented by Miller and Wegman [15] and Welch [21] and seem to work ....

A. Lempel and J. Ziv, "On the complexity of finite sequences," IEEE Trans. Inform. Theory, vol. IT-22, pp. 75-81, Jan. 1976.

....Research Triangle Institute, RTP, NC 27709. 707 708 lnllJJ IMAGE Ire I TROFUI ENCODER . IMAGE image I TRANSFORM [ ooefficienlsl TABLE F . r . lCOU.Esso. DECODER Fig. 1. A typical data compression system. arithmetic coding [6] or Lempel Ziv type methods [7,8] to attain a lossless technique that exceeds the performance of stand alone lossless methods. 1.1 Vector quantization Vector quantization is a lossy block coding technique that is used extensively to compress data at low bit rates (high compression ratios) Vector quantizers compress image data ....

Lempel, A.; Ziv, J. On the complexity of finite sequences. IEEE Trans. on Information Theory, 22:75-81; 1976.

....1, 82. The size of each U i was not less that 50 100 thousand letters. Let us consider lossless data compression algorithms. The following algorithms have been most popular recently: Hu man coding, arithmetic coding, Burrows Wheeler technique [17] and many variations of Lempel Zip coding [18]. Some algorithms are specially aimed on text coding: these are PPM [19] this algorithm uses Markov model of small order) and DMC [20] this algorithm uses dynamical Markov Coding) Each algorithm has a huge number of variations and parameters (for example, there exist so called dynamic Hu man ....

Lempel A., Ziv J. On the Complexity of Finite Sequences, IEEE Trans. on Inform. Theory., 1976, vol. 22, no. 1, pp. 75-81.

....should use at most O(n) extra space, optimally O(m) in addition to the n length compressed file. The first compressed pattern matching algorithms dealt with Ziv Lempel compressed text. In [Farach and Thorup 1995] was presented a compressed matching algorithm for the LZ1 classic compression scheme [Ziv and Lempel 1976] that runs in O(n log 2 (u=n) m) time. In [Amir et al. 1996] a compressed matching algorithm for the LZ78 compression scheme was presented, which finds the first occurrence in O(n m 2 ) time and space, or in O(n log m m) time and in O(n m) space. An extension of [Amir et al. 1996] to ....

Ziv, J. and Lempel, A. 1976. On the complexity of finite sequences. IEEE Transactions on Information Theory 22, 75--81.

.... of compression methods based on the idea of self reference: while the textfile is scanned, substrings or phrases are identified and stored in a dictionary, and whenever, later in the process, a phrase or concatenation of phrases is encountered again, this is compactly encoded by suitable pointers [97, 138, 139]. Of the several existing versions of the method, we describe below the one known as Lempel Ziv Welsh method, which is incarnated by the compress feature under the UNIX operating system. For the encoding, a dictionary is initialized with all the characters of the alphabet. At the generic ....

A. Lempel and J. Ziv. On the complexity of finite sequences. IEEE Trans. on information Theory, 22:75--81, 1976.

.... of strings [55] finding the longest substring that appears in h out of k strings, for any h 2 [58] computing characteristic strings [59] matching a string as an arbitrary path of an unrooted labeled tree [4] performing efficient dictionary matching [6, 5, 7, 21, 43] data compression schemes [39, 40, 73, 83, 84, 102, 103]; searching for the longest run of a given motif in molecular sequences [53, 54, 100] metric distance on strings [34] complexity measure on random strings for cryptology [81] inverted indices [22] analyzing genetic sequences [25, 23] finding duplication in programming code [13] generating ....

....the pathstring y having locus in the parent of leaf j, and append to y the (j jyj) th character of x , say a. Thus y is a prefix of x[j; n] that occurs at least twice in x, but ya is a prefix of x[j; n] that occurs only once. Problems of this kind are found also in data compression schemes [39, 73, 83, 84, 102, 103], in compressing assembly code [40] and in searching for the longest run of a given motif in molecular sequences [53, 54, 100] Suffix trees help also to design elegant algorithms for finding squares [10, 68] and repetitions in a string [10] computing statistics for the non overlapping ....

Lempel, A., and Ziv, A., On the complexity of finite sequences, IEEE Trans. Information Theory, 22, 75--81, (1976).

....agency for research on human genome, GREG. E mail addresses: stephane.grumbach inria.fr, fariza.tahi inria.fr. 1 codewords are used for (non) frequent patterns [Huf52] In the second one, factors of different lengths are encoded using a pointer to one of their previous occurrences in the text [LZ76] Unfortunately, the compression of genetic sequences happens to be a very difficult task. They are at first glance very similar to random strings, and have only very hidden regularities. The classical algorithms for text compression fail to compress genetic sequences. Worse, they may extend the ....

A. Lempel and J. Ziv. On the complexity of finite sequences. IEEE Trans. Inform. Theory, 22(1):75--81, 1976.

.... one, blocks of fixed length (generally letters) are encoded with respect to their probability of appearance; short (long) codewords are used for (non) frequent patterns [Huf52] In the second one, factors of different length are encoded using a pointer to their previous occurrences in the text [LZ76] Unfortunately, the compression of DNA sequences appears to be a difficult task. They are at first glance very similar to random strings, and have only very hidden regularities. The classical algorithms for text compression do not work on DNA sequences. Worse, they extend the content of the ....

A. Lempel and J. Ziv. On the complexity of finite sequences. IEEE Trans. Inform. Theory, 22(1):75--81, 1976.

....1 1. INTRODUCTION Sux trees have found a wide variety of applications in algorithms on words including: the longest repeated substring [16] squares or repetitions in strings [1] string statistics [1] string matching [4] approximate string matching [4] string comparison, compression schemes [9], implementation of Lempel Ziv algorithm, genetic sequences, biologically signi cant motif patterns in DNA [4] sequence assembly [4] approximate overlaps [4] and so forth. It is fair to say that sux trees are most widely used data structure in algorithms on words. Despite this, very little is ....

A. Lempel and J. Ziv. On the Complexity of Finite Sequences. IEEE Information Theory, 22:75-81, 1976.

.... of a given sequence X (cf. 34] In data compression schemes, the following problem is of prime interest: for a given data base subsequence of length n, find the longest prefix of the (n 1)st suffix S n 1 which is not a prefix of any other suffixes S i (1 i n) of the data base sequence (cf. [25], 43] 44] And last but not least, in molecular sequences comparison (e.g. finding homology between DNA sequences) one may search for the longest run of a given motif (cf. 16] 17] 40] a unique sequence or the longest alignment (cf. 13] 40] These, and several other problems on ....

.... of applications in algorithms on words including: the longest repeated substring (cf. 41] squares or repetitions in strings (cf. 3] string statistics (cf. 3] string matching (cf. 9] 42] approximate string matching (cf. 12] 9] 42] string comparison, compression schemes (cf. [25]) implementation of the Lempel Ziv algorithm, genetic sequences, biologically significant motif patterns in DNA (cf. 9] 40] sequence assembly (cf. 9] approximate overlaps (cf. 9] 40] and so forth. It is fair to say that suffix trees are the most widely used data structure in ....

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A. Lempel and J. Ziv, On the Complexity of Finite Sequences, IEEE Information Theory 22, 1, 75-81 (1976).

....characterized the patternlessness of a finite sequence by the length of the shortest Turing machine program that could generate the sequence. This concept was further developed by Martin Loef [48] A different approach to evaluating the complexity of finite sequences was given by Lempel and Ziv [41]. They further employed their definition for the universal data compression [83] A measure for the complexity of a finite sequence S is given in terms of the number of new patterns which appear as we move along the sequence [43] This number is C(S) called the sequence complexity of S. Example ....

A. Lempel and J. Ziv, "On the Complexity of Finite Sequences", IEEE Trans. on IT, Vol.22, No.1, pp.75--81, Jan., 1976.

....2 We are now ready to present the proof of Theorem 3.4. Proof of Theorem 3.4: It has been shown in [ZiLb] that Compression E;n (oe n 1 ) c(oe n 1 ) 1 n lg(2ff(c(oe n 1 ) 1) 3.8) where c(oe n 1 ) is the maximum number of nodes in any parse tree 1 for oe n 1 . It is shown in [LeZ] that 0 c(oe n 1 ) n lg ff (1 Gamma ffl n ) lg n ; lim n 1 ffl n = 0: 3.9) Theorem 3.4 is clearly true when c(oe n 1 ) o(n= lg n) since Compression E;n (oe n 1 ) 0 as n 1. When c(oe n 1 ) O(n= lg n) using the lower bound for Compression M(s) n (oe n 1 ) from Lemma 3.3 ....

A. Lempel and J. Ziv, "On the Complexity of Finite Sequences," IEEE Transactions on Information Theory 22 (January 1976).

....Rate 11 We are now ready to present the proof of Theorem 4. Proof of Theorem 4: It has been shown in [ZiL] that Compression E;n (oe n 1 ) c(oe n 1 ) 1 n lg(2ff(c(oe n 1 ) 1) 7) where c(oe n 1 ) is the maximum number of nodes in any parse tree 1 for oe n 1 . It is shown in [LeZ] that 0 c(oe n 1 ) n lg ff (1 Gamma ffl n ) lg n ; where lim n 1 ffl n = 0: 8) Theorem 4 is clearly true when c(oe n 1 ) o(n= lg n) since Compression E;n (oe n 1 ) 0 as n 1. When c(oe n 1 ) O(n= lg n) using the lower bound for Compression M(s) n (oe n 1 ) from Lemma 3 ....

A. Lempel & J. Ziv, "On the Complexity of Finite Sequences," IEEE Transactions on Information Theory 22 (January 1976).

....validity. The first measure, based on the work of Tanguiane (1993) uses the idea that a temporal pattern can be described in terms of (elaborations of) more simple patterns, simultaneously at different levels. The second measure is based on the complexity measure for finite sequences proposed by Lempel and Ziv (1976), which is related to the number of steps in a self delimiting production process by which such a sequence is presumed to be generated. The third measure, newly developed here, is rooted in the theoretical framework of rhythm perception of Povel and Essens (1985) It takes into account the ease of ....

....The first measure is based on the work of Tanguiane (1993) and uses the idea that a rhythmic pattern can be described in terms of (elaborations of) more simple patterns, simultaneously at different levels. The second measure is based on the complexity measure for finite sequences proposed by Lempel and Ziv (1976), which is related to the number of steps in a self delimiting production process by which such a sequence is presumed to be generated. Finally, the third measure proposed is rooted in the theoretical framework of rhythm perception discussed in Povel and Essens (1985) This measure takes into ....

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Lempel, A. & Ziv, J. (1976). On the complexity of finite sequences. IEEE Transactions on Information Theory. IT-22 (1).

....the Lempel Ziv grammar transform and the bisection grammar transform. 15 3.1.1 Lempel Ziv Grammar Transform Let x = x 1 x 2 Delta Delta Delta x n be an A string. Let (u 1 ; u 2 ; u t ) be the Lempel Ziv parsing of x, by which we mean the parsing of x established in the paper [15] and used in the 1978 version of the Lempel Ziv data compression algorithm [27] Let S lz (x) be the set of substrings of x defined by S lz (x) Delta = fxg [ fu 1 ; u 2 ; u t g For each u 2 S lz (x) let (s u ; a u ) be the parsing of u in which a u 2 A, and let A u be a variable ....

.... A 00 0 A 001 A 00 1 A 0 0 A 1 1 The grammar [G lz x ] can be verified by the reader to be the grammar in G (f0; 1g) with the production rules A 0 A 1 A 2 A 3 A 4 A 5 A 6 A 1 0 A 2 1 A 3 A 1 0 A 4 A 2 0 A 5 A 3 0 A 6 A 3 1 Discussion. The reader of [15] will find notions called producibility and reproducibility introduced there that allow one to describe a recursive copying process for certain parsings of a data string (not just the parsing considered above) For each such parsing, it is easy to construct a grammar which embodies this copying ....

A. Lempel and J. Ziv, "On the Complexity of Finite Sequences," IEEE Trans. Inform. Theory, vol. 22, pp. 75--81, 1976.

....Richter Award, Brandeis University [GilbertK 92] Our purpose here is to give a lower bound on that factor for an infinite class of sequences, the bound growing as the logarithm of sequence length. In the process we also get a reversal lower bound for the original Lempel Ziv complexity measure [LempelZ 76]. We then improve this separation bound to a polynomial factor, subject only to a condition that has been found true for the particular constructions we employ, and we argue why the condition should be dispensible. 2 Complexities of Sequences and their Reversals Each sequence in the family we ....

....would be just i were it not for the possibility of that a tuple first occurs straddling a boundary a relatively rare event. Even so, deleting the 2 would not change the asymptotic result. The complexity measure defined by Lempel and Ziv in their 1976 paper was called by the authors C LZ [LempelZ 76]. This measure is defined by parsing a string according to the rule repeatedly parse the longest substring which can be found to begin earlier in the string here the new substring need not have been an earlier parsed phrase. The authors showed this measure to be a lower bound on any parsing of ....

A. Lempel and J. Ziv "On the Complexity of Finite Sequences ", IEEE Transactions on Information Theory 22:1, 75-81. (1976)

....0 p(x) 1. This normalized quantity p(x) that depends solely on x will be referred to as the (finite state) compressibility of x. In Theorem 1 (the converse to coding theorem) we derive a lower bound on pe( xD and show that in the limit this bound approaches the normalized Lempel Ziv complexity [4] and becomes a lower bound on the compressibility of x. In Theorem 2 (the coding theorem) we demonstrate, using a variant of the author s universal compression algorithm [5] the existence of an asymptoti cally optimal universal ILF encoding scheme under which the compression ratio attained for ....

A. Lempel and J. Ziv, "On the complexity of finite sequences," IEEE Trans. Inforr Theory, vol. IT-22, pp. 75-81, Jan. 1976.

....Haifa, Israel. He is now with the Sperry Research Center, Sudbury, MA 01776. then show that the efficiency of our universal code with no a priori knowledge of the source approaches those bounds. The proposed compression algorithm is an adaptation of a simple copying procedure discussed recently [10] in a study on the complexity of finite sequences. Basically, we employ the concept of encoding future segments of the source output via maximum length copying from a buffer containing the recent past output. The transmitted codeword consists of the buffer address and the length of the copied ....

A Lempel and J. Ziv, "On the complexity of finite sequences," IEEE Trans. Inform. Theory, vol. IT-22, pp. 75-81, Jan. 1976.

....9 54 9 54 1 6 1 6 1 6 2 3 2 3 2 3 1 6 1 6 1 6 Figure 1. Parse tree for alphabet extension. 83 33 54 21 6 12 8 6 4 4 9 aaa b c ab ac aab aac 16 9 17 Figure 2. Hu man code tree for Leaves of Parse Tree. 8. 2 LZ76: precursor to parsing In their 1976 paper [LZ76], the basic technique is to parse a sequence into phrases, and following a phrase with a real or imaginary comma as a delimiter. Thus popular subsequences of symbols are treated as a unit. Both the algorithms LZ77 and LZ78 operate using a dictionary of based on the decoded data. The coding ....

A. Lempel and J. Ziv, \On the Complexity of Finite Sequences", IEEE Trans. on Info. Theory, vol 22, 75 - 81, 1976.

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A. Lempel and J. Ziv, On the complexity of finite sequences. IEEE Trans. Inform. Theory, vol. IT-22, no. 1, pp. 75--81, January 1976.

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A. Lempel and J. Ziv, On the Complexity of finite sequences, IEEE Trans. on Inf. Theory 22 (1976) pp. 75--81.

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J. ZIV AND A. LEMPEL, "On the complexity of finite sequences", IEEE Transactions on Information Theory, 22 (1976), 75--81.

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A. Lempel and J. Ziv, On the complexity of finite sequences, IEEE Trans. on Inf. Theory 22, 75-81 (1976).

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A. Lempel and J.Ziv, On the complexity of finite sequences, IEEE Trans. on Inf. Theory 22, 75-81 (1976)

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