Performance of LZ Algorithms on Individual Sequences

The performance of three versions of the Lempel-Ziv (1976) algorithm on individual sequences is investigated. It is shown that as the restart length goes to infinity, each compresses an individual sequence as well as any block-to-variable finite-state information lossless algorithm, and that the same conclusion holds for sliding-window LZ as the window width goes to infinity. Examples are given showing that an infinite-memory version outperforms such finite-memory forms and that such finite-memory forms can compress more than the Ziv (1978) entropy, which is the best compression attainable by finite-state block-to-block codes that have vanishing probability of error.

[1]  de Ng Dick Bruijn.  A combinatorial problem , 1946 .

[2]  Richard A. Games,et al.  On the Complexities of de Bruijn Sequences , 1982, J. Comb. Theory, Ser. A.

[3]  Abraham Lempel,et al.  On the Complexity of Finite Sequences , 1976, IEEE Trans. Inf. Theory.

[4]  P. Landrock,et al.  The Number of Cross-Join Pairs in Maximum Length Linear Sequences , 1991 .

[5]  Fred S. Annexstein.  Generating De Bruijn Sequences: An Efficient Implementation , 1997, IEEE Trans. Computers.

[6]  Abraham Lempel,et al.  On a Homomorphism of the de Bruijn Graph and its Applications to the Design of Feedback Shift Registers , 1970, IEEE Transactions on Computers.

[7]  Jacob Ziv,et al.  Coding theorems for individual sequences , 1978, IEEE Trans. Inf. Theory.

[8]  Serap A. Savari,et al.  Redundancy of the Lempel-Ziv String Matching Code , 1998, IEEE Trans. Inf. Theory.

[9]  Terry A. Welch,et al.  A Technique for High-Performance Data Compression , 1984, Computer.

[10]  Tuvi Etzion.  The depth distribution-a new characterization for linear codes , 1997, IEEE Trans. Inf. Theory.

[11]  Kenneth G. Paterson,et al.  Near optimal single-track Gray codes , 1996, IEEE Trans. Inf. Theory.

[12]  Abraham Lempel,et al.  A universal algorithm for sequential data compression , 1977, IEEE Trans. Inf. Theory.

[13]  Serap A. Savari,et al.  Redundancy of the Lempel-Ziv incremental parsing rule , 1997, IEEE Trans. Inf. Theory.

[14]  Abraham Lempel,et al.  Cryptology in Transition , 1979, CSUR.

[15]  A. D. Wyner,et al.  The sliding-window Lempel-Ziv algorithm is asymptotically optimal , 1994, Proc. IEEE.

[16]  Iickho Song,et al.  Cross-Joins in de Bruijn Sequences and Maximum Length Linear Sequences^* , 1993 .

[17]  Kenneth G. Paterson,et al.  A method for constructing decodable de Bruijn sequences , 1996, IEEE Trans. Inf. Theory.

[18]  Abraham Lempel,et al.  Compression of individual sequences via variable-rate coding , 1978, IEEE Trans. Inf. Theory.