Optimal Maximal Prefix Coding and Huffman Coding

Huffman coding has been widely used in data, image, and video compression. Novel maximal prefix coding different from the Huffman coding is introduced. Relationships between the Huffman coding and optimal maximal prefix coding are discussed. We show that all Huffman coding schemes are maximal prefix coding schemes and have the shortest average code word length among maximal prefix coding schemes. Moreover, it is proven that, for any maximal prefix code C, there exists a suitable information source I = ( , P) such that C is exactly a Huffman code for I = ( , P). So it is essential to show that the class of Huffman codes is coincident with the class of maximal prefix codes. A case study of applying to data compression is also given. Comparing with the Huffman coding, maximal prefix coding is a more flexible compression method.

[1]  J. Berstel,et al.  Theory of codes , 1985 .

[2]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[3]  Jeffrey Scott Vitter,et al.  Design and analysis of dynamic Huffman codes , 1987, JACM.

[4]  Shawmin Lei,et al.  An entropy coding system for digital HDTV applications , 1991, IEEE Trans. Circuits Syst. Video Technol..

[5]  Ian H. Witten,et al.  Text Compression , 1990, 125 Problems in Text Algorithms.

[6]  Marcel Paul Schützenberger,et al.  On the Synchronizing Properties of Certain Prefix Codes , 1964, Inf. Control..

[7]  Stephanie Perkins,et al.  Binary Huffman Equivalent Codes with a Short Synchronizing Codeword , 1998, IEEE Trans. Inf. Theory.

[8]  Joan L. Mitchell,et al.  JPEG: Still Image Data Compression Standard , 1992 .

[9]  Darrel Hankerson,et al.  Introduction to Information Theory and Data Compression , 2003 .

[10]  Sihan Qing,et al.  Information Security for Global Information Infrastructures , 2000, IFIP — The International Federation for Information Processing.

[11]  David A. Huffman,et al.  A method for the construction of minimum-redundancy codes , 1952, Proceedings of the IRE.

[12]  Brockway McMillan,et al.  Two inequalities implied by unique decipherability , 1956, IRE Trans. Inf. Theory.

[13]  Marcel Paul Schützenberger,et al.  On Synchronizing Prefix Codes , 1967, Inf. Control..

[14]  Steven Roman.  Introduction to coding and information theory , 1997, Undergraduate texts in mathematics.

[15]  Mark Nelson,et al.  The Data Compression Book , 2009 .

[16]  H. Shyr.  Free monoids and languages , 1979 .

[17]  Dongyang Long,et al.  On Group Codes , 1996, Theor. Comput. Sci..

[18]  Kou-Hu Tzou.  High-order entropy coding for images , 1992, IEEE Trans. Circuits Syst. Video Technol..