Advertisement

Correspondences Between Variable Length Parsing and Coding Problems

  • Julia Abrahams
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 107)

Abstract

Stubley and Blake’s minimum discrimination parse tree problem is seen to correspond to Karp’s variable length unequal costs coding problem for arbitrary source distributions in the sense that, where Stubley and Blake minimize the discrimination D(P ∥ Q),Karp minimizes D(Q ∥ P). In the special case that Q is uniform, Stubley and Blake’s problem is Tunstall parsing and Karp’s problem is Varn coding. In the special case that P is dyadic, Stubley and Blake’s problem is of interest because Karp’s problem is Huffman coding.

Similarly, Lempel, Even, and Cohn’s parse tree problem can be interpreted as minimizing G(PQ) for a particular functional G in the special case that P is dyadic. The problem of minimizing G(P ∥ Q) for arbitrary P is seen to correspond to Karp’s problem in the sense that Karp also minimizes G(Q ∥ P). In the special case that Q is uniform, Tunstall parsing and Varn coding correspond to each other, and in the special case that P is dyadic, Lempel, Even, Cohn parsing and Huffman coding correspond to each other.

Key words

Variable-length-to-block source coding block-to-variable-length source coding unequal costs coding parse tree Stubley and Blake algorithm Lempel Even and Cohn algorithm Karp algorithm Tunstall algorithm Varn algorithm Huffman algorithm 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    F. Fabris, “Variable-length-to-variable-length source coding: a greedy step-by-step algorithm,” IEEE Trans. Inform. Theory, vol. 38, pp. 1609–1617, Sep. 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    G.H. Freeman, “Divergence and the construction of variable-to-variable-length lossless codes by source-word extensions,” Proc., Data Compression Conference, Mar. 30-Apr. 2, 1993, Snowbird, UT, J.A. Storer and M. Cohn, eds., IEEE Computer Soc. Press, Los Alamitos, CA, pp. 79–88, 1993.Google Scholar
  3. [3]
    M.J. golin and G. Rote, “A dynamic programming algorithm for constructing optimal prefix-free codes for unequal letter costs,” Proc., ICALP’95, 22nd Intl. Colloq. on Automata, Languages, and Programming, Jul. 10–14, 1995; Szeged, Hungary, Z. Fulop and F. Gecseg, eds., Lecture Notes in Computer Science, vol. 944, Springer Verlag, NY, pp. 256–267, 1995.Google Scholar
  4. [4]
    D.A. Huffman, “A method for the construction of minimum redundancy codes,” Proc. IRE, vol. 40, pp. 1098–1101, 1952.CrossRefGoogle Scholar
  5. [5]
    F. Jelinek and G. Longo, “Algorithms for source coding,” Coding and, Complexity, G. Longo, ed., International Centre for Mechanical Sciences-Courses and Lectures, no. 216, Springer Verlag, NY, pp. 293–330, 1975.Google Scholar
  6. [6]
    R.M. Karp, “Minimum-redundancy coding for the discrete noiseless channel,” IRE Trans. Inform. Theory, vol. IT-7, pp. 27–38, Jan. 1961.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    A. Lempel, S. Even, and M. Cohn, “An algorithm for optimal prefix parsing of a noiseless and memoryless channel,” IEEE Trans. Inform. Theory, vol. IT-19, pp. 208–214, Mar. 1973.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    S.A. Savari, “Some notes on Varn coding,” IEEE Trans Inform. Theory, vol. 40, pp. 181–186, Jan. 1994.zbMATHCrossRefGoogle Scholar
  9. [9]
    P.R. Stubley and I.F. Blake, “On a discrete probability distribution matching problem,” Preprint, Nov. 16, 1992.Google Scholar
  10. [10]
    P.R. Stubley, “Adaptive variable-to-variable length codes,” Proc., Data Compression Conference, Mar. 29–31, 1994, Snowbird, UT, J.A. Storer and M. Cohn, eds., IEEE Computer Soc. Press, Los Alamitos, CA, pp. 98–105, 1994.Google Scholar
  11. [11]
    B.P. TUNSTALL, “Synthesis of noiseless compression codes,” Ph.D. Dissertation, Georgia Inst. Tech., Atlanta, GA, 1968.Google Scholar
  12. [12]
    B.F. Varn, “Optimal variable length codes (arbitrary symbol cost and equal code word probabilities),” Inform. Contr., vol. 19, pp. 289–301, 1971.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Julia Abrahams
    • 1
  1. 1.Mathematical, Computer, and Information Sciences DivisionOffice of Naval Re­searchArlingtonUSA

Personalised recommendations