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Data Compression

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Introduction   ::   Kraft   ::   Optimal Codes   ::   Bounds   ::   Huffman   ::   Coding

Huffman codes

Huffman codes are the optimal prefix codes for a given distribution. What's more, if we know the distribution, Huffman codes are easy to find. The code operates from the premise of assigning longer codewords to less-likely symbols, and doing it in a tree-structured way so that the codes obtained are prefix-free.
\begin{example} Consider the distribution $X$\ taking values in the set $\Xc = ... ... assign codewords on the tree. The average codelength is 2.3 bits. \end{example}

Codes with more than D=2 symbols can also be built, as described in the book.

Proving the optimality of the Huffman code begins with the following simple lemma:
\begin{lemma} For any distribution, there exists an optimal instantaneous code ... ...it, and correspond to the two least-likely symbols. \end{enumerate}\end{lemma}

\begin{proof}(Sketch) \begin{enumerate} \item Simply swap lengths. \item If t... ...st code, which contradicts the optimality property. \end{enumerate}\end{proof}

For a code on m symbols, assume (w.o.l.o.g.) that the probabilities are ordered $p_1> p_2 > \cdots > p_m$ p_2 > \cdots > p_m$" align="middle" border="0" height="28" width="136" />. Define the merged code on m-1 symbols by merging the two least probable symbols pm, pm-1. The codeword on this merged symbol is the common prefix on the two least-probable (longest) codewords, which, by the lemma, exists. The expected length of the code Cm is

\begin{displaymath}\begin{aligned} L(C_m) &= \sum_{i=1}^m p_i l_i \\ &= \sum_{i... ...+ p_m(l_{m}+1) \\ &= L(C_{m-1}) + p_m(l_{m}'+1). \end{aligned}\end{displaymath}

The optimization problem on m symbols has been reduced to an optimization problem on m-1 symbols. Proceeding inductively, we get down to two symbols, for which the optimal code is obvious: 0 or 1.

Copyright 2008, by the Contributing Authors. Cite/attribute Resource. admin. (2006, May 17). Data Compression. Retrieved March 21, 2010, from Free Online Course Materials — USU OpenCourseWare Web site: http://ocw.usu.edu/Electrical_and_Computer_Engineering/Information_Theory/lecture6_6.htm. This work is licensed under a Creative Commons License. Creative Commons License
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