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<title>Adaptive Huffman coding</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Adaptive Huffman coding</span></span>

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<p><b>Adaptive Huffman coding</b> (also called <b>Dynamic Huffman coding</b>) is an <a href="Adaptive_coding" title="Adaptive coding">adaptive coding</a> technique based on <a href="Huffman_coding" title="Huffman coding">Huffman coding</a>. It permits building the code as the symbols are being transmitted, having no initial knowledge of source distribution, that allows one-pass encoding and adaptation to changing conditions in data.<sup id="cite_ref-LiDrew2014_1-0" class="reference"><a href="#cite_note-LiDrew2014-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>The benefit of one-pass procedure is that the source can be encoded in real time, though it becomes more sensitive to transmission errors, since just a single loss ruins the whole code, requiring <a href="Error_detection_and_correction" title="Error detection and correction">error detection and correction</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Algorithms">Algorithms</h2></div>

<p>There are a number of implementations of this method, the most notable are <b>FGK</b> (<a href="Newton_Faller" title="Newton Faller">Faller</a>-<a href="Robert_G._Gallager" title="Robert G. Gallager">Gallager</a>-<a href="Donald_Knuth" title="Donald Knuth">Knuth</a>) and <b><a href="Jeffrey_Vitter" title="Jeffrey Vitter">Vitter</a></b> algorithm.

</p>

<div class="mw-heading mw-heading3"><h3 id="FGK_Algorithm">FGK Algorithm</h3></div>

<p>It is an online coding technique based on Huffman coding. Having no initial knowledge of occurrence frequencies, it permits dynamically adjusting the Huffman's tree as data are being transmitted. In a FGK Huffman tree, a special external node, called <i>0-node</i>, is used to identify a newly coming character. That is, whenever new data is encountered, output the path to the 0-node followed by the data. For a past-coming character, just output the path of the data in the current Huffman's tree. Most importantly, we have to adjust the FGK Huffman tree if necessary, and finally update the frequency of related nodes. As the frequency of a datum is increased, the<i> sibling property</i> of the Huffman's tree may be broken. The adjustment is triggered for this reason. It is accomplished by consecutive swappings of nodes, subtrees, or both. The data node is swapped with the highest-ordered node of the same frequency in the Huffman's tree, (or the subtree rooted at the highest-ordered node). All ancestor nodes of the node should also be processed in the same manner.

</p><p>Since the FGK Algorithm has some drawbacks about the node-or-subtree swapping, Vitter proposed another algorithm to improve it.

</p>

<div class="mw-heading mw-heading3"><h3 id="Vitter_algorithm">Vitter algorithm</h3></div>

<p>Some important terminologies & constraints :-

</p>

<ul><li><b>Implicit Numbering</b> : It simply means that nodes are numbered in increasing order by level and from left to right. i.e. nodes at bottom level will have low implicit number as compared to upper level nodes and nodes on same level are numbered in increasing order from left to right. In other terms, when we have built the Huffman tree, when merging two nodes into a parent node, we have set the one with the lower value as the left child, and the one with the higher value as the right child.</li>

<li><b>Invariant</b> : For each weight w, all leaves of weight w precede all internal nodes having weight w. In other terms, when we have built the Huffman tree, if several nodes had the same value, we prioritized merging the leaves over the internal nodes.</li>

<li><b>Blocks</b> : Nodes of same weight and same type (i.e. either leaf node or internal node) form a Block.</li>

<li><b>Leader</b> : Highest numbered node in a block.</li></ul>

<p>Blocks are interlinked by increasing order of their weights.

</p><p>A leaf block always precedes internal block of same weight, thus maintaining the invariant.

</p><p><b>NYT (Not Yet Transferred)</b> is a special node used to represent symbols which are <i>'not yet transferred'</i>.

</p>

<pre><b>algorithm</b> for adding a symbol <b>is</b>

leaf_to_increment := NULL

p := pointer to the leaf node containing the next symbol

Extend p by adding two children

Left child becomes new NYT and right child is the new symbol leaf node

<i>p</i> := parent of new symbol leaf node

leaf_to_increment := Right Child of p

Swap p with leader of its block

<b>if</b> (new p is sibling to NYT) <b>then</b>

leaf_to_increment := p

<i>p</i> := parent of p

<b>while</b> (p ≠ NULL) <b>do</b>

Slide_And_Increment(p)

<b>if</b> (leaf_to_increment != NULL) <b>then</b>

Slide_And_Increment(leaf_to_increment)

</pre>

<pre><b>function</b> Slide_And_Increment(p) <b>is</b>

previous_p := parent of <i>p</i>

<b>if</b> (p is an internal node) <b>then</b>

Slide p in the tree higher than the leaf nodes of weight wt + 1

increase weight of <i>p</i> by 1

<i>p</i> := previous_p

Slide p in the tree higher than the internal nodes of weight wt

increase weight of <i>p</i> by 1

<i>p</i> := new parent of <i>p</i>.

</pre>

<p>Encoder and decoder start with only the root node, which has the maximum number. In the beginning it is our initial NYT node.

</p><p>When we transmit an NYT symbol, we have to transmit code for the NYT node, then for its generic code.

</p><p>For every symbol that is already in the tree, we only have to transmit code for its leaf node.

</p>

<div class="mw-heading mw-heading4"><h4 id="Example">Example</h4></div>

</p><p>Encoding "abb" gives 01100001 001100010 11.

</p><p>Step 1:

</p><p>Start with an empty tree.

</p><p>For "a" transmit its binary code.

</p><p>Step 2:

</p><p>NYT spawns two child nodes: 254 and 255, both with weight 0. Increase weight for root and 255. Code for "a", associated with node 255, is 1.

</p><p>For "b" transmit 0 (for NYT node) then its binary code.

</p><p>Step 3:

</p><p>NYT spawns two child nodes: 252 for NYT and 253 for leaf node, both with weight 0. Increase weights for 253, 254, and root. To maintain Vitter's invariant that all leaves of weight w precede (in the implicit numbering) all internal nodes of weight w, the branch starting with node 254 should be swapped (in terms of symbols and weights, but not number ordering) with node 255. Code for "b" is 11.

</p><p>For the second "b" transmit 11.

</p><p>For the convenience of explanation this step doesn't exactly follow Vitter's algorithm,<sup id="cite_ref-:0_2-0" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> but the effects are equivalent.

</p><p>Step 4:

</p><p>Go to leaf node 253. Notice we have two blocks with weight 1. Node 253 and 254 is one block (consisting of leaves), node 255 is another block (consisting of internal nodes). For node 253, the biggest number in its block is 254, so swap the weights and symbols of nodes 253 and 254. Now node 254 and the branch starting from node 255 satisfy the SlideAndIncrement condition<sup id="cite_ref-:0_2-1" class="reference"><a href="#cite_note-:0-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> and hence must be swapped. At last increase node 255 and 256's weight.

</p><p>Future code for "b" is 1, and for "a" is now 01, which reflects their frequency.

</p>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFZe-Nian_LiMark_S._DrewJiangchuan_Liu2014" class="citation book cs1">Ze-Nian Li; Mark S. Drew; Jiangchuan Liu (9 April 2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=R6vBBAAAQBAJ"><i>Fundamentals of Multimedia</i></a>. Springer Science & Business Media. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-319-05290-8</bdi>.</cite></span>

</li>

<li id="cite_note-:0-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.cs.duke.edu/csed/curious/compression/adaptivehuff.html#tree">"Adaptive Huffman Coding"</a>. Cs.duke.edu<span class="reference-accessdate">. Retrieved <span class="nowrap">2012-02-26</span></span>.</cite></span>

</li>

</ol></div></div>

<ul><li>Vitter's original paper: J. S. Vitter, "<a rel="nofollow" class="external text" href="http://www.ittc.ku.edu/~jsv/Papers/Vit87.jacmACMversion.pdf">Design and Analysis of Dynamic Huffman Codes</a>", Journal of the ACM, 34(4), October 1987, pp 825–845.</li>

<li>J. S. Vitter, "ALGORITHM 673 Dynamic Huffman Coding", ACM Transactions on Mathematical Software, 15(2), June 1989, pp 158–167. Also appears in Collected Algorithms of ACM.</li>

<li>Donald E. Knuth, "Dynamic Huffman Coding", Journal of Algorithm, 6(2), 1985, pp 163–180.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><span class="noviewer" typeof="mw:File"><span></span></span> This article incorporates <a href="Copyright_status_of_works_by_the_federal_government_of_the_United_States" title="Copyright status of works by the federal government of the United States">public domain material</a> from <cite id="CITEREFPaul_E._Black" class="citation cs1">Paul E. Black. <a rel="nofollow" class="external text" href="https://xlinux.nist.gov/dads/HTML/adaptiveHuffman.html">"adaptive Huffman coding"</a>. <i><a href="Dictionary_of_Algorithms_and_Data_Structures" class="mw-redirect" title="Dictionary of Algorithms and Data Structures">Dictionary of Algorithms and Data Structures</a></i>. <a href="NIST" class="mw-redirect" title="NIST">NIST</a>.</cite></li>

<li><a rel="nofollow" class="external text" href="http://www.ics.uci.edu/~dan/pubs/DC-Sec4.html">University of California Dan Hirschberg site</a></li>

<li><a rel="nofollow" class="external text" href="http://www.cs.cf.ac.uk/Dave/Multimedia/node212.html">Cardiff University Dr. David Marshall site</a></li>

<li><a rel="nofollow" class="external text" href="https://code.google.com/p/compression-code/downloads/list">C implementation of Vitter algorithm</a></li>

<li><a rel="nofollow" class="external text" href="http://www.cs.duke.edu/csed/curious/compression/adaptivehuff.html">Excellent description from Duke University</a></li></ul>

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</style><div id="Data_compression_methods241" style="font-size:114%;margin:0 4em"><a href="Data_compression" title="Data compression">Data compression</a> methods</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Lossless_compression" title="Lossless compression">Lossless</a><br>type</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Entropy_coding" title="Entropy coding">Entropy</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Adaptive_coding" title="Adaptive coding">Adaptive coding</a></li>

<li><a href="Arithmetic_coding" title="Arithmetic coding">Arithmetic</a></li>

<li><a href="Asymmetric_numeral_systems" title="Asymmetric numeral systems">Asymmetric numeral systems</a></li>

<li><a href="Golomb_coding" title="Golomb coding">Golomb</a></li>

<li><a href="Huffman_coding" title="Huffman coding">Huffman</a>

<ul>

<li><a href="Canonical_Huffman_code" title="Canonical Huffman code">Canonical</a></li>

<li><a href="Modified_Huffman_coding" title="Modified Huffman coding">Modified</a></li></ul></li>

<li><a href="Range_coding" title="Range coding">Range</a></li>

<li><a href="Shannon_coding" title="Shannon coding">Shannon</a></li>

<li><a href="Shannon%E2%80%93Fano_coding" title="Shannon–Fano coding">Shannon–Fano</a></li>

<li><a href="Shannon%E2%80%93Fano%E2%80%93Elias_coding" title="Shannon–Fano–Elias coding">Shannon–Fano–Elias</a></li>

<li><a href="Tunstall_coding" title="Tunstall coding">Tunstall</a></li>

<li><a href="Unary_coding" title="Unary coding">Unary</a></li>

<li><a href="Universal_code_(data_compression)" title="Universal code (data compression)">Universal</a>

<ul><li><a href="Exponential-Golomb_coding" title="Exponential-Golomb coding">Exp-Golomb</a></li>

<li><a href="Fibonacci_coding" title="Fibonacci coding">Fibonacci</a></li>

<li><a href="Elias_gamma_coding" title="Elias gamma coding">Gamma</a></li>

<li><a href="Levenshtein_coding" title="Levenshtein coding">Levenshtein</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Dictionary_coder" title="Dictionary coder">Dictionary</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Byte-pair_encoding" title="Byte-pair encoding">Byte-pair encoding</a></li>

<li><a href="LZ77_and_LZ78" title="LZ77 and LZ78">Lempel–Ziv</a>

<li><a href="Snappy_(compression)" title="Snappy (compression)">Snappy</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Delta_encoding" title="Delta encoding">Delta</a>

<ul><li><a href="Incremental_encoding" title="Incremental encoding">Incremental</a></li></ul></li>

<li><a href="Grammar-based_code" title="Grammar-based code">Grammar</a>

<li><a href="Sequitur_algorithm" title="Sequitur algorithm">Sequitur</a></li></ul></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Hybrid</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li>LZ77 + Huffman

<ul><li><a href="Deflate" title="Deflate">Deflate</a></li>

<li>LZ77 + ANS

<ul><li><a href="LZFSE" title="LZFSE">LZFSE</a></li></ul></li>

<li>LZ77 + Huffman + ANS

<ul><li><a href="Zstd" title="Zstd">Zstandard</a></li></ul></li>

<li>LZ77 + Huffman + context

<ul><li><a href="Brotli" title="Brotli">Brotli</a></li></ul></li>

<li>LZSS + Huffman

<li>LZ77 + Range

<li>LZHAM</li></ul></li>

<li>RLE + BWT + MTF + Huffman

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Lossy_compression" title="Lossy compression">Lossy</a><br>type</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Transform_coding" title="Transform coding">Transform</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Discrete_cosine_transform" title="Discrete cosine transform">Discrete cosine transform</a>

<li><a href="Wavelet_transform" title="Wavelet transform">Wavelet</a>

<ul><li><a href="Daubechies_wavelet" title="Daubechies wavelet">Daubechies</a></li>

<li><a href="Set_partitioning_in_hierarchical_trees" title="Set partitioning in hierarchical trees">SPIHT</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Predictive</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Adaptive_differential_pulse-code_modulation" title="Adaptive differential pulse-code modulation">ADPCM</a></li></ul></li>

<ul><li><a href="Algebraic_code-excited_linear_prediction" title="Algebraic code-excited linear prediction">ACELP</a></li>

<li>Motion

<ul><li><a href="Motion_compensation" title="Motion compensation">Compensation</a></li>

<li><a href="Motion_estimation" title="Motion estimation">Estimation</a></li>

<li><a href="Motion_vector" class="mw-redirect" title="Motion vector">Vector</a></li></ul></li>

<li><a href="Psychoacoustics" title="Psychoacoustics">Psychoacoustic</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Data_compression#Audio" title="Data compression">Audio</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Companding" title="Companding">Companding</a></li>

<li><a href="Convolution" title="Convolution">Convolution</a></li>

<li><a href="Dynamic_range" title="Dynamic range">Dynamic range</a></li>

<li><a href="Latency_(audio)" title="Latency (audio)">Latency</a></li>

<li><a href="Nyquist%E2%80%93Shannon_sampling_theorem" title="Nyquist–Shannon sampling theorem">Nyquist–Shannon theorem</a></li>

<li><a href="Sampling_(signal_processing)" title="Sampling (signal processing)">Sampling</a></li>

<li><a href="Silence_compression" title="Silence compression">Silence compression</a></li>

<li><a href="Sound_quality" title="Sound quality">Sound quality</a></li>

<li><a href="Speech_coding" title="Speech coding">Speech coding</a></li>

<li><a href="Sub-band_coding" title="Sub-band coding">Sub-band coding</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Audio_codec" title="Audio codec">Codec</a><br>parts</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Adaptive_differential_pulse-code_modulation" title="Adaptive differential pulse-code modulation">ADPCM</a></li>

<ul><li><a href="Algebraic_code-excited_linear_prediction" title="Algebraic code-excited linear prediction">ACELP</a></li>

<li><a href="Psychoacoustics" title="Psychoacoustics">Psychoacoustic model</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Image_compression" title="Image compression">Image</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Chroma_subsampling" title="Chroma subsampling">Chroma subsampling</a></li>

<li><a href="Coding_tree_unit" title="Coding tree unit">Coding tree unit</a></li>

<li><a href="Color_space" title="Color space">Color space</a></li>

<li><a href="Compression_artifact" title="Compression artifact">Compression artifact</a></li>

<li><a href="Image_resolution" title="Image resolution">Image resolution</a></li>

<li><a href="Macroblock" title="Macroblock">Macroblock</a></li>

<li><a href="Pixel" title="Pixel">Pixel</a></li>

<li><a href="Quantization_(image_processing)" title="Quantization (image processing)">Quantization</a></li>

<li><a href="Standard_test_image" title="Standard test image">Standard test image</a></li>

<li><a href="Texture_compression" title="Texture compression">Texture compression</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Methods</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Chain_code" title="Chain code">Chain code</a></li>

<li><a href="Deflate" title="Deflate">Deflate</a></li>

<li><a href="Fractal_compression" title="Fractal compression">Fractal</a></li>

<li><a href="Wavelet_transform" title="Wavelet transform">Wavelet</a>

<ul><li><a href="Daubechies_wavelet" title="Daubechies wavelet">Daubechies</a></li>

<li><a href="Set_partitioning_in_hierarchical_trees" title="Set partitioning in hierarchical trees">SPIHT</a></li></ul></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Data_compression#Video" title="Data compression">Video</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Display_resolution" title="Display resolution">Display resolution</a></li>

<li><a href="Film_frame" title="Film frame">Frame</a></li>

<li><a href="Frame_rate" title="Frame rate">Frame rate</a></li>

<li><a href="Video_compression_picture_types" title="Video compression picture types">Frame types</a></li>

<li><a href="Interlaced_video" title="Interlaced video">Interlace</a></li>

<li><a href="Video#Characteristics_of_video_streams" title="Video">Video characteristics</a></li>

<li><a href="Video_quality" title="Video quality">Video quality</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Video_codec" title="Video codec">Codec</a><br>parts</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Deblocking_filter" title="Deblocking filter">Deblocking filter</a></li>

<li><a href="Lapped_transform" title="Lapped transform">Lapped transform</a></li>

<li>Motion

<ul><li><a href="Motion_compensation" title="Motion compensation">Compensation</a></li>

<li><a href="Motion_estimation" title="Motion estimation">Estimation</a></li>

<li><a href="Motion_vector" class="mw-redirect" title="Motion vector">Vector</a></li></ul></li>

<li><a href="Wavelet_transform" title="Wavelet transform">Wavelet</a>

<ul><li><a href="Daubechies_wavelet" title="Daubechies wavelet">Daubechies</a></li>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Information_theory" title="Information theory">Theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Compressed_data_structure" title="Compressed data structure">Compressed data structures</a>

<ul><li><a href="Compressed_suffix_array" title="Compressed suffix array">Compressed suffix array</a></li>

<li><a href="FM-index" title="FM-index">FM-index</a></li></ul></li>

<li><a href="Entropy_(information_theory)" title="Entropy (information theory)">Entropy</a></li>

<li><a href="Information_theory" title="Information theory">Information theory</a>

<ul><li><a href="Timeline_of_information_theory" title="Timeline of information theory">Timeline</a></li></ul></li>

<li><a href="Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a></li>

<li><a href="Prefix_code" title="Prefix code">Prefix code</a></li>

<li><a href="Quantization_(signal_processing)" title="Quantization (signal processing)">Quantization</a></li>

<li><a href="Rate%E2%80%93distortion_theory" title="Rate–distortion theory">Rate–distortion</a></li>

<li><a href="Redundancy_(information_theory)" title="Redundancy (information theory)">Redundancy</a></li>

<li><a href="Data_compression_symmetry" title="Data compression symmetry">Symmetry</a></li>

<li><a href="Smallest_grammar_problem" title="Smallest grammar problem">Smallest grammar problem</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Community</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hutter_Prize" title="Hutter Prize">Hutter Prize</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">People</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Mark_Adler" title="Mark Adler">Mark Adler</a></li>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2024-12-05" href="https://en.wikipedia.org/wiki/?title=Adaptive_Huffman_coding&oldid=1261412240">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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