2516:
determine the character value of that particular leaf. The process continues recursively until the last leaf node is reached; at that point, the
Huffman tree will thus be faithfully reconstructed. The overhead using such a method ranges from roughly 2 to 320 bytes (assuming an 8-bit alphabet). Many other techniques are possible as well. In any case, since the compressed data can include unused "trailing bits" the decompressor must be able to determine when to stop producing output. This can be accomplished by either transmitting the length of the decompressed data along with the compression model or by defining a special code symbol to signify the end of input (the latter method can adversely affect code length optimality, however).
2474:
particular byte value). Before this can take place, however, the
Huffman tree must be somehow reconstructed. In the simplest case, where character frequencies are fairly predictable, the tree can be preconstructed (and even statistically adjusted on each compression cycle) and thus reused every time, at the expense of at least some measure of compression efficiency. Otherwise, the information to reconstruct the tree must be sent a priori. A naive approach might be to prepend the frequency count of each character to the compression stream. Unfortunately, the overhead in such a case could amount to several kilobytes, so this method has little practical use. If the data is compressed using
2038:
5159:
5149:
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1992:, the entropy is a measure of the smallest codeword length that is theoretically possible for the given alphabet with associated weights. In this example, the weighted average codeword length is 2.25 bits per symbol, only slightly larger than the calculated entropy of 2.205 bits per symbol. So not only is this code optimal in the sense that no other feasible code performs better, but it is very close to the theoretical limit established by Shannon.
359:
20:
456:
2989:, where a 'dash' takes longer to send than a 'dot', and therefore the cost of a dash in transmission time is higher. The goal is still to minimize the weighted average codeword length, but it is no longer sufficient just to minimize the number of symbols used by the message. No algorithm is known to solve this in the same manner or with the same efficiency as conventional Huffman coding, though it has been solved by
2168:. A binary tree is generated from left to right taking the two least probable symbols and putting them together to form another equivalent symbol having a probability that equals the sum of the two symbols. The process is repeated until there is just one symbol. The tree can then be read backwards, from right to left, assigning different bits to different branches. The final Huffman code is:
345:(sometimes called "prefix-free codes", that is, the bit string representing some particular symbol is never a prefix of the bit string representing any other symbol). Huffman coding is such a widespread method for creating prefix codes that the term "Huffman code" is widely used as a synonym for "prefix code" even when such a code is not produced by Huffman's algorithm.
2591:
approaches the entropy limit, i.e., optimal compression. However, blocking arbitrarily large groups of symbols is impractical, as the complexity of a
Huffman code is linear in the number of possibilities to be encoded, a number that is exponential in the size of a block. This limits the amount of blocking that is done in practice.
2587:. Prefix codes, and thus Huffman coding in particular, tend to have inefficiency on small alphabets, where probabilities often fall between these optimal (dyadic) points. The worst case for Huffman coding can happen when the probability of the most likely symbol far exceeds 2 = 0.5, making the upper limit of inefficiency unbounded.
3334:
If weights corresponding to the alphabetically ordered inputs are in numerical order, the
Huffman code has the same lengths as the optimal alphabetic code, which can be found from calculating these lengths, rendering HuâTucker coding unnecessary. The code resulting from numerically (re-)ordered input
2709:
involves calculating the probabilities dynamically based on recent actual frequencies in the sequence of source symbols, and changing the coding tree structure to match the updated probability estimates. It is used rarely in practice, since the cost of updating the tree makes it slower than optimized
2590:
There are two related approaches for getting around this particular inefficiency while still using
Huffman coding. Combining a fixed number of symbols together ("blocking") often increases (and never decreases) compression. As the size of the block approaches infinity, Huffman coding theoretically
2559:
Although both aforementioned methods can combine an arbitrary number of symbols for more efficient coding and generally adapt to the actual input statistics, arithmetic coding does so without significantly increasing its computational or algorithmic complexities (though the simplest version is slower
2515:
is the number of bits per symbol). Another method is to simply prepend the
Huffman tree, bit by bit, to the output stream. For example, assuming that the value of 0 represents a parent node and 1 a leaf node, whenever the latter is encountered the tree building routine simply reads the next 8 bits to
23:
Huffman tree generated from the exact frequencies of the text "this is an example of a huffman tree". Encoding the sentence with this code requires 135 (or 147) bits, as opposed to 288 (or 180) bits if 36 characters of 8 (or 5) bits were used (This assumes that the code tree structure is known to the
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and
Huffman coding produce equivalent results — achieving entropy — when every symbol has a probability of the form 1/2. In other circumstances, arithmetic coding can offer better compression than Huffman coding because — intuitively — its "code words" can have effectively
2618:
Many variations of
Huffman coding exist, some of which use a Huffman-like algorithm, and others of which find optimal prefix codes (while, for example, putting different restrictions on the output). Note that, in the latter case, the method need not be Huffman-like, and, indeed, need not even be
2331:
The process begins with the leaf nodes containing the probabilities of the symbol they represent. Then, the process takes the two nodes with smallest probability, and creates a new internal node having these two nodes as children. The weight of the new node is set to the sum of the weight of the
2473:
Generally speaking, the process of decompression is simply a matter of translating the stream of prefix codes to individual byte values, usually by traversing the
Huffman tree node by node as each bit is read from the input stream (reaching a leaf node necessarily terminates the search for that
2464:
It is generally beneficial to minimize the variance of codeword length. For example, a communication buffer receiving
Huffman-encoded data may need to be larger to deal with especially long symbols if the tree is especially unbalanced. To minimize variance, simply break ties between queues by
2043:
BED". In steps 2 to 6, the letters are sorted by increasing frequency, and the least frequent two at each step are combined and reinserted into the list, and a partial tree is constructed. The final tree in step 6 is traversed to generate the dictionary in step 7. Step 8 uses it to encode the
2564:
issues. Thus many technologies have historically avoided arithmetic coding in favor of Huffman and other prefix coding techniques. As of mid-2010, the most commonly used techniques for this alternative to Huffman coding have passed into the public domain as the early patents have expired.
3467:
only optimally matches a symbol of probability 1/2 and other probabilities are not represented optimally; whereas the code word length in arithmetic coding can be made to exactly match the true probability of the symbol. This difference is especially striking for small alphabet sizes.
1915:
2403:, the first one containing the initial weights (along with pointers to the associated leaves), and combined weights (along with pointers to the trees) being put in the back of the second queue. This assures that the lowest weight is always kept at the front of one of the two queues:
2560:
and more complex than Huffman coding). Such flexibility is especially useful when input probabilities are not precisely known or vary significantly within the stream. However, Huffman coding is usually faster and arithmetic coding was historically a subject of some concern over
2543:
Huffman's original algorithm is optimal for a symbol-by-symbol coding with a known input probability distribution, i.e., separately encoding unrelated symbols in such a data stream. However, it is not optimal when the symbol-by-symbol restriction is dropped, or when the
2217:
of the source is 1.74 bits/symbol. If this Huffman code is used to represent the signal, then the average length is lowered to 1.85 bits/symbol; it is still far from the theoretical limit because the probabilities of the symbols are different from negative powers of
1054:
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enables one to use any kind of weights (costs, frequencies, pairs of weights, non-numerical weights) and one of many combining methods (not just addition). Such algorithms can solve other minimization problems, such as minimizing
761:
1703:
317:
on the problem of finding the most efficient binary code. Huffman, unable to prove any codes were the most efficient, was about to give up and start studying for the final when he hit upon the idea of using a frequency-sorted
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children. We then apply the process again, on the new internal node and on the remaining nodes (i.e., we exclude the two leaf nodes), we repeat this process until only one node remains, which is the root of the Huffman tree.
1148:
3001:
In the standard Huffman coding problem, it is assumed that any codeword can correspond to any input symbol. In the alphabetic version, the alphabetic order of inputs and outputs must be identical. Thus, for example,
2465:
choosing the item in the first queue. This modification will retain the mathematical optimality of the Huffman coding while both minimizing variance and minimizing the length of the longest character code.
2978:, no matter how many of those digits are 0s, how many are 1s, etc. When working under this assumption, minimizing the total cost of the message and minimizing the total number of digits are the same thing.
2984:
is the generalization without this assumption: the letters of the encoding alphabet may have non-uniform lengths, due to characteristics of the transmission medium. An example is the encoding alphabet of
2524:
The probabilities used can be generic ones for the application domain that are based on average experience, or they can be the actual frequencies found in the text being compressed. This requires that a
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followed by the use of prefix codes; these are often called "Huffman codes" even though most applications use pre-defined variable-length codes rather than codes designed using Huffman's algorithm.
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here is the number of symbols in the alphabet, which is typically a very small number (compared to the length of the message to be encoded); whereas complexity analysis concerns the behavior when
2722:
Most often, the weights used in implementations of Huffman coding represent numeric probabilities, but the algorithm given above does not require this; it requires only that the weights form a
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is a variant where the goal is still to achieve a minimum weighted path length, but there is an additional restriction that the length of each codeword must be less than a given constant. The
1983:
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coding. This reflects the fact that compression is not possible with such an input, no matter what the compression method, i.e., doing nothing to the data is the optimal thing to do.
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In the standard Huffman coding problem, it is assumed that each symbol in the set that the code words are constructed from has an equal cost to transmit: a code word whose length is
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2166:
3449:
3395:
1564:, the sum of the probability budgets across all symbols is always less than or equal to one. In this example, the sum is strictly equal to one; as a result, the code is termed a
1568:
code. If this is not the case, one can always derive an equivalent code by adding extra symbols (with associated null probabilities), to make the code complete while keeping it
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Huffman coding is optimal among all methods in any case where each input symbol is a known independent and identically distributed random variable having a probability that is
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methods, more common symbols are generally represented using fewer bits than less common symbols. Huffman's method can be efficiently implemented, finding a code in time
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24:
decoder and thus does not need to be counted as part of the transmitted information). The frequencies and codes of each character are shown in the accompanying table.
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to 1 contractor; for binary coding, this is a 2 to 1 contractor, and any sized set can form such a contractor. If the number of source words is congruent to 1 modulo
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solution to this optimal binary alphabetic problem, which has some similarities to Huffman algorithm, but is not a variation of this algorithm. A later method, the
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878:
2921:
2022:
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2326:
1995:
In general, a Huffman code need not be unique. Thus the set of Huffman codes for a given probability distribution is a non-empty subset of the codes minimizing
262:
table for encoding a source symbol (such as a character in a file). The algorithm derives this table from the estimated probability or frequency of occurrence (
2891:
2598:. This technique adds one step in advance of entropy coding, specifically counting (runs) of repeated symbols, which are then encoded. For the simple case of
2300:
2248:
1074:
597:
675:
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non-integer bit lengths, whereas code words in prefix codes such as Huffman codes can only have an integer number of bits. Therefore, a code word of length
2606:
is optimal among prefix codes for coding run length, a fact proved via the techniques of Huffman coding. A similar approach is taken by fax machines using
2529:
must be stored with the compressed text. See the Decompression section above for more information about the various techniques employed for this purpose.
2024:
for that probability distribution. (However, for each minimizing codeword length assignment, there exists at least one Huffman code with those lengths.)
2419:
Create a new internal node, with the two just-removed nodes as children (either node can be either child) and the sum of their weights as the new weight.
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node. As a common convention, bit '0' represents following the left child and bit '1' represents following the right child. A finished tree has up to
1079:
3316:(1977), uses simpler logic to perform the same comparisons in the same total time bound. These optimal alphabetic binary trees are often used as
274:
to the number of input weights if these weights are sorted. However, although optimal among methods encoding symbols separately, Huffman coding
1910:{\displaystyle H(A)=\sum _{w_{i}>0}w_{i}h(a_{i})=\sum _{w_{i}>0}w_{i}\log _{2}{1 \over w_{i}}=-\sum _{w_{i}>0}w_{i}\log _{2}{w_{i}}.}
3983:
2450:
The final encoding of any symbol is then read by a concatenation of the labels on the edges along the path from the root node to the symbol.
2410:
Enqueue all leaf nodes into the first queue (by probability in increasing order so that the least likely item is in the head of the queue).
1189:
We give an example of the result of Huffman coding for a code with five characters and given weights. We will not verify that it minimizes
2687:-ary tree for Huffman coding. In these cases, additional 0-probability place holders must be added. This is because the tree must form an
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4462:
2431:
Once the Huffman tree has been generated, it is traversed to generate a dictionary which maps the symbols to binary codes as follows:
4351:
5198:
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and is often the code used in practice, due to ease of encoding/decoding. The technique for finding this code is sometimes called
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Create a new internal node with these two nodes as children and with probability equal to the sum of the two nodes' probabilities.
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to develop a similar code. Building the tree from the bottom up guaranteed optimality, unlike the top-down approach of
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1989:
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2649:-ary tree. This approach was considered by Huffman in his original paper. The same algorithm applies as for binary (
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5082:
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883:
3671:
Gallager, R.G.; van Voorhis, D.C. (1975). "Optimal source codes for geometrically distributed integer alphabets".
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511:
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4700:
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376:
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Abrahams, J. (1997-06-11). "Code and parse trees for lossless source encoding". Written at Arlington, VA, USA.
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409:
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2041:
Visualisation of the use of Huffman coding to encode the message "A_DEAD_DAD_CEDED_A_BAD_BABE_A_BEADED_ABACA_
473:
4827:
3005:
2610:. However, run-length coding is not as adaptable to as many input types as other compression technologies.
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5055:
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node which makes it easy to read the code (in reverse) starting from a leaf node. Internal nodes contain a
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391:
3404:
3350:
2894:
1049:{\textstyle L\left(C\left(W\right)\right)=\sum _{i=1}^{n}{w_{i}\operatorname {length} \left(c_{i}\right)}}
4145:
4140:
3472:
5193:
5087:
5014:
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283:
3797:"A Dynamic Programming Algorithm for Constructing Optimal Prefix-Free Codes with Unequal Letter Costs"
341:
Huffman coding uses a specific method for choosing the representation for each symbol, resulting in a
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4964:
4812:
4402:
4397:
4252:
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236:
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2538:
672:, which is the tuple of the (positive) symbol weights (usually proportional to probabilities), i.e.
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internal nodes. A Huffman tree that omits unused symbols produces the most optimal code lengths.
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330:
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2926:
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For a set of symbols with a uniform probability distribution and a number of members which is a
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4203:
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3713:
3528:
3329:
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least probable symbols are taken together, instead of just the 2 least probable. Note that for
2475:
2400:
255:, and published in the 1952 paper "A Method for the Construction of Minimum-Redundancy Codes".
4940:
5203:
5072:
3951:
3650:
Gribov, Alexander (2017-04-10). "Optimal Compression of a Polyline with Segments and Arcs".
3630:
3397:, which, having the same codeword lengths as the original solution, is also optimal. But in
2453:
In many cases, time complexity is not very important in the choice of algorithm here, since
2213:
The standard way to represent a signal made of 4 symbols is by using 2 bits/symbol, but the
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4180:
4001:
3911:
2844:
1985:. So for simplicity, symbols with zero probability can be left out of the formula above.)
856:
259:
3471:
Prefix codes nevertheless remain in wide use because of their simplicity, high speed, and
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1998:
8:
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4878:
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4695:
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1059:
756:{\displaystyle w_{i}=\operatorname {weight} \left(a_{i}\right),\,i\in \{1,2,\dots ,n\}}
582:
302:
224:
4641:
3343:, since it is optimal like Huffman coding, but alphabetic in weight probability, like
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4842:
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time, unlike the presorted and unsorted conventional Huffman problems, respectively.
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whose solution has been refined for the case of integer costs by Mordecai J. Golin.
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is the maximum length of a codeword. No algorithm is known to solve this problem in
2416:
Dequeue the two nodes with the lowest weight by examining the fronts of both queues.
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The use of asymmetric numeral systems as an accurate replacement for Huffman coding
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approach very similar to that used by Huffman's algorithm. Its time complexity is
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1920:(Note: A symbol with zero probability has zero contribution to the entropy, since
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1198:
3850:(1971). "Optimal Computer Search Trees and Variable-Length Alphabetical Codes".
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1143:{\displaystyle L\left(C\left(W\right)\right)\leq L\left(T\left(W\right)\right)}
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Proceedings. Compression and Complexity of SEQUENCES 1997 (Cat. No.97TB100171)
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Remove the two nodes of highest priority (lowest probability) from the queue
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2552:, a single code may be insufficient for optimality. Other methods such as
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3888:(1998), "Algorithm G (GarsiaâWachs algorithm for optimum binary trees)",
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2695:−1, then the set of source words will form a proper Huffman tree.
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3871:
3631:"Profile: David A. Huffman: Encoding the "Neatness" of Ones and Zeroes"
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314:
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3475:. They are often used as a "back-end" to other compression methods.
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2427:
The remaining node is the root node; the tree has now been generated.
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Create a leaf node for each symbol and add it to the priority queue.
2266:(frequency of appearance) of the symbol and optionally, a link to a
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greater than 2, not all sets of source words can properly form an
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where the node with lowest probability is given highest priority:
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4240:
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2478:, the compression model can be precisely reconstructed with just
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19:
3704:. Division of Mathematics, Computer & Information Sciences,
3347:. The HuffmanâShannonâFano code corresponding to the example is
2446:. Repeat the process at both the left child and the right child.
2438:
If node is not a leaf node, label the edge to the left child as
4331:
2561:
3890:
The Art of Computer Programming, Vol. 3: Sorting and Searching
3765:"Minimum-redundancy coding for the discrete noiseless channel"
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4311:
3484:
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2826:
Length-limited Huffman coding/minimum variance Huffman coding
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2364:
Since efficient priority queue data structures require O(log
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The remaining node is the root node and the tree is complete.
266:) for each possible value of the source symbol. As in other
305:
classmates were given the choice of a term paper or a final
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4160:
4150:
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3521:"A Method for the Construction of Minimum-Redundancy Codes"
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of the given set of weights; the result is nearly optimal.
306:
846:{\displaystyle C\left(W\right)=(c_{1},c_{2},\dots ,c_{n})}
4296:
4262:
3492:
2258:. Initially, all nodes are leaf nodes, which contain the
299:
3244:{\displaystyle H\left(A,C\right)=\left\{0,10,11\right\}}
3177:{\displaystyle H\left(A,C\right)=\left\{00,01,1\right\}}
3110:{\displaystyle H\left(A,C\right)=\left\{00,1,01\right\}}
3587:
Ze-Nian Li; Mark S. Drew; Jiangchuan Liu (2014-04-09).
2422:
Enqueue the new node into the rear of the second queue.
258:
The output from Huffman's algorithm can be viewed as a
2729:, meaning a way to order weights and to add them. The
2230:, the size of which depends on the number of symbols,
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325:
In doing so, Huffman outdid Fano, who had worked with
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2714:, which is more flexible and has better compression.
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If the symbols are sorted by probability, there is a
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2001:
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among all compression methods - it is replaced with
3947:
Huffman coding in various languages on Rosetta Code
3928:, Second Edition. MIT Press and McGraw-Hill, 2001.
3670:
490:codeword length (equivalently, a tree with minimum
383:. Unsourced material may be challenged and removed.
322:and quickly proved this method the most efficient.
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3294:
3243:
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2997:Optimal alphabetic binary trees (HuâTucker coding)
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2865:
2814:
2667:
2503:
2320:
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2016:
1977:
1909:
1675:(in bits) is the weighted sum, across all symbols
1658:{\displaystyle h(a_{i})=\log _{2}{1 \over w_{i}}.}
1657:
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1142:
1068:
1048:
936:
872:
853:, which is the tuple of (binary) codewords, where
845:
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591:
571:
3894:. See also History and bibliography, pp. 453â454.
2413:While there is more than one node in the queues:
239:. The process of finding or using such a code is
5180:
3892:(2nd ed.), AddisonâWesley, pp. 451â453
3263:, the authors of the paper presenting the first
2742:
2572:, Huffman coding is equivalent to simple binary
2346:While there is more than one node in the queue:
1928:
2594:A practical alternative, in widespread use, is
2407:Start with as many leaves as there are symbols.
1978:{\displaystyle \lim _{w\to 0^{+}}w\log _{2}w=0}
3614:J. Duda, K. Tahboub, N. J. Gadil, E. J. Delp,
3977:
3323:
2822:, a problem first applied to circuit design.
2717:
2645:â 1} alphabet to encode message and build an
2399:)) method to create a Huffman tree using two
1697:, of the information content of each symbol:
937:{\displaystyle a_{i},\,i\in \{1,2,\dots ,n\}}
3991:
3438:
3408:
3384:
3354:
2155:
2131:
2111:
2059:
931:
907:
750:
726:
665:{\displaystyle W=(w_{1},w_{2},\dots ,w_{n})}
572:{\displaystyle A=(a_{1},a_{2},\dots ,a_{n})}
472:A set of symbols and their weights (usually
3554:
2335:The simplest construction algorithm uses a
2226:of nodes. These can be stored in a regular
2117:{\displaystyle \{a_{1},a_{2},a_{3},a_{4}\}}
286:if a better compression ratio is required.
3984:
3970:
2698:
2556:often have better compression capability.
3717:
3655:
3593:. Springer Science & Business Media.
3580:
900:
719:
498:
443:Learn how and when to remove this message
3842:
3699:
3117:, but instead should be assigned either
2982:Huffman coding with unequal letter costs
2966:Huffman coding with unequal letter costs
2435:Start with current node set to the root.
2047:
2036:
454:
18:
3810:(5) (published 1998-09-01): 1770â1781.
3804:IEEE Transactions on Information Theory
3674:IEEE Transactions on Information Theory
3628:
3515:
2626:
2550:independent and identically distributed
2548:are unknown. Also, if symbols are not
2539:Arithmetic coding § Huffman coding
2376:â1 nodes, this algorithm operates in O(
2052:A source generates 4 different symbols
579:, which is the symbol alphabet of size
463:
5181:
3770:IRE Transactions on Information Theory
3649:
3607:
3560:"On the construction of Huffman trees"
3043:{\displaystyle A=\left\{a,b,c\right\}}
2368:) time per insertion, and a tree with
3965:
3952:Huffman codes (python implementation)
3884:
3794:
2161:{\displaystyle \{0.4;0.35;0.2;0.05\}}
348:
3758:
3444:{\displaystyle \{110,111,00,01,10\}}
3390:{\displaystyle \{000,001,01,10,11\}}
1372:Contribution to weighted path length
1193:over all codes, but we will compute
1056:be the weighted path length of code
381:adding citations to reliable sources
352:
3852:SIAM Journal on Applied Mathematics
3795:Golin, Mordekai J. (January 1998).
2442:and the edge to the right child as
13:
2974:digits will always have a cost of
2837:solves this problem with a simple
2785:
2782:
2779:
2776:
2773:
2770:
2519:
2222:The technique works by creating a
2027:
492:weighted path length from the root
486:(a set of codewords) with minimum
14:
5215:
3957:A visualization of Huffman coding
3940:
3936:. Section 16.3, pp. 385â392.
3619:, Picture Coding Symposium, 2015.
5158:
5157:
5148:
5147:
2468:
357:
231:is a particular type of optimal
5199:Lossless compression algorithms
3901:
3878:
3836:
3788:
3454:
1990:Shannon's source coding theorem
1174:{\displaystyle T\left(W\right)}
368:needs additional citations for
275:
3752:
3693:
3664:
3643:
3622:
3548:
3509:
3289:
3274:
2948:
2933:
2910:
2904:
2860:
2851:
2815:{\displaystyle \max _{i}\left}
2641:algorithm uses the {0, 1,...,
2355:Add the new node to the queue.
2032:
2011:
2005:
1935:
1771:
1758:
1716:
1710:
1616:
1603:
840:
795:
659:
614:
566:
521:
336:
1:
3502:
3483:'s algorithm) and multimedia
2831:Length-limited Huffman coding
2613:
2532:
1590:with non-null probability is
1457:Information content (in bits)
3251:. This is also known as the
2504:{\displaystyle B\cdot 2^{B}}
1540:
1537:
1534:
1531:
1528:
1525:
1491:
1488:
1485:
1482:
1479:
1476:
1451:
1448:
1445:
1442:
1439:
1436:
1412:
1409:
1406:
1403:
1400:
1397:
1366:
1363:
1360:
1357:
1354:
1328:
1323:
1318:
1313:
1308:
1283:
1280:
1277:
1274:
1271:
1268:
1246:
1243:
1240:
1237:
1234:
243:, an algorithm developed by
7:
3495:have a front-end model and
3341:HuffmanâShannonâFano coding
3050:could not be assigned code
2511:bits of information (where
1558:, meaning that the code is
459:Constructing a Huffman Tree
10:
5220:
5039:Compressed data structures
4361:RLE + BWT + MTF + Huffman
4029:Asymmetric numeral systems
3925:Introduction to Algorithms
3728:10.1109/SEQUEN.1997.666911
3590:Fundamentals of Multimedia
3542:10.1109/JRPROC.1952.273898
3327:
3324:The canonical Huffman code
3295:{\displaystyle O(n\log n)}
2954:{\displaystyle O(n\log n)}
2731:Huffman template algorithm
2718:Huffman template algorithm
2712:adaptive arithmetic coding
2546:probability mass functions
2536:
2388:is the number of symbols.
2278:and an optional link to a
1686:with non-zero probability
1579:, the information content
1184:
289:
284:asymmetric numeral systems
235:that is commonly used for
16:Technique to compress data
5143:
5127:
5111:
5029:
4954:
4886:
4877:
4800:
4734:
4725:
4626:
4543:
4534:
4450:
4398:Discrete cosine transform
4388:
4379:
4328:LZ77 + Huffman + context
4281:
4191:
4121:
4009:
4000:
2675:) codes, except that the
2250:. A node can be either a
1583:(in bits) of each symbol
1424:
1339:Codeword length (in bits)
1333:
1288:
1209:
237:lossless data compression
5103:Smallest grammar problem
3782:10.1109/TIT.1961.1057615
3706:Office of Naval Research
3687:10.1109/TIT.1975.1055357
3335:is sometimes called the
2461:grows to be very large.
5044:Compressed suffix array
4593:NyquistâShannon theorem
3473:lack of patent coverage
2835:package-merge algorithm
2706:adaptive Huffman coding
2699:Adaptive Huffman coding
2608:modified Huffman coding
1497:Contribution to entropy
484:prefix-free binary code
3529:Proceedings of the IRE
3445:
3399:canonical Huffman code
3391:
3337:canonical Huffman code
3330:Canonical Huffman code
3306:GarsiaâWachs algorithm
3296:
3245:
3178:
3111:
3044:
2955:
2917:
2887:
2867:
2816:
2669:
2505:
2322:
2296:
2244:
2219:
2162:
2118:
2045:
2018:
1979:
1911:
1659:
1197:and compare it to the
1175:
1144:
1070:
1050:
1009:
938:
874:
847:
757:
666:
593:
573:
499:Formalized description
460:
216:
5073:Kolmogorov complexity
4941:Video characteristics
4318:LZ77 + Huffman + ANS
3629:Huffman, Ken (1991).
3446:
3392:
3297:
3246:
3179:
3112:
3045:
2956:
2918:
2888:
2868:
2866:{\displaystyle O(nL)}
2817:
2670:
2506:
2323:
2297:
2245:
2163:
2119:
2051:
2040:
2019:
1980:
1912:
1660:
1554:For any code that is
1176:
1145:
1071:
1051:
989:
939:
875:
873:{\displaystyle c_{i}}
848:
758:
667:
594:
574:
458:
276:is not always optimal
22:
5163:Compression software
4757:Compression artifact
4713:Psychoacoustic model
3912:Charles E. Leiserson
3712:. pp. 145â171.
3405:
3351:
3268:
3188:
3121:
3054:
3006:
2927:
2916:{\displaystyle O(n)}
2898:
2877:
2845:
2738:
2653:
2482:
2306:
2286:
2234:
2128:
2056:
2017:{\displaystyle L(C)}
1999:
1988:As a consequence of
1924:
1704:
1597:
1154:
1080:
1060:
956:
884:
880:is the codeword for
857:
775:
676:
605:
583:
512:
464:Informal description
377:improve this article
260:variable-length code
5153:Compression formats
4792:Texture compression
4787:Standard test image
4603:Silence compression
3636:Scientific American
3345:ShannonâFano coding
3318:binary search trees
2703:A variation called
2668:{\displaystyle n=2}
2630:-ary Huffman coding
2600:Bernoulli processes
2596:run-length encoding
2321:{\displaystyle n-1}
1561:uniquely decodeable
331:ShannonâFano coding
5061:Information theory
4916:Display resolution
4742:Chroma subsampling
4131:Byte pair encoding
4076:ShannonâFanoâElias
3776:(1). IEEE: 27â38.
3441:
3387:
3292:
3241:
3174:
3107:
3040:
2951:
2913:
2883:
2863:
2812:
2750:
2727:commutative monoid
2665:
2501:
2476:canonical encoding
2318:
2292:
2240:
2220:
2158:
2114:
2046:
2014:
1975:
1949:
1907:
1868:
1799:
1744:
1655:
1428:Probability budget
1171:
1140:
1066:
1046:
934:
870:
843:
753:
662:
589:
569:
476:to probabilities).
461:
349:Problem definition
303:information theory
225:information theory
217:
5194:1952 in computing
5176:
5175:
5025:
5024:
4975:Deblocking filter
4873:
4872:
4721:
4720:
4530:
4529:
4375:
4374:
3816:10.1109/18.705558
3784:– via IEEE.
3600:978-3-319-05290-8
3460:Arithmetic coding
3314:Michelle L. Wachs
2886:{\displaystyle L}
2741:
2554:arithmetic coding
2295:{\displaystyle n}
2243:{\displaystyle n}
2212:
2211:
2124:with probability
1927:
1846:
1838:
1777:
1722:
1650:
1552:
1551:
1069:{\displaystyle C}
592:{\displaystyle n}
453:
452:
445:
427:
309:. The professor,
280:arithmetic coding
215:
214:
5211:
5189:Data compression
5161:
5160:
5151:
5150:
4980:Lapped transform
4884:
4883:
4762:Image resolution
4747:Coding tree unit
4732:
4731:
4541:
4540:
4386:
4385:
4007:
4006:
3993:Data compression
3986:
3979:
3972:
3963:
3962:
3916:Ronald L. Rivest
3908:Thomas H. Cormen
3895:
3893:
3886:Knuth, Donald E.
3882:
3876:
3875:
3840:
3834:
3833:
3831:
3830:
3801:
3792:
3786:
3785:
3767:
3760:Karp, Richard M.
3756:
3750:
3749:
3721:
3708:(ONR). Salerno:
3697:
3691:
3690:
3668:
3662:
3661:
3659:
3647:
3641:
3640:
3626:
3620:
3611:
3605:
3604:
3584:
3578:
3577:
3575:
3574:
3564:
3556:Van Leeuwen, Jan
3552:
3546:
3545:
3536:(9): 1098â1101.
3525:
3513:
3450:
3448:
3447:
3442:
3401:, the result is
3396:
3394:
3393:
3388:
3301:
3299:
3298:
3293:
3250:
3248:
3247:
3242:
3240:
3236:
3212:
3208:
3183:
3181:
3180:
3175:
3173:
3169:
3145:
3141:
3116:
3114:
3113:
3108:
3106:
3102:
3078:
3074:
3049:
3047:
3046:
3041:
3039:
3035:
2960:
2958:
2957:
2952:
2922:
2920:
2919:
2914:
2892:
2890:
2889:
2884:
2872:
2870:
2869:
2864:
2821:
2819:
2818:
2813:
2811:
2807:
2806:
2802:
2801:
2788:
2765:
2764:
2749:
2674:
2672:
2671:
2666:
2514:
2510:
2508:
2507:
2502:
2500:
2499:
2327:
2325:
2324:
2319:
2301:
2299:
2298:
2293:
2249:
2247:
2246:
2241:
2170:
2169:
2167:
2165:
2164:
2159:
2123:
2121:
2120:
2115:
2110:
2109:
2097:
2096:
2084:
2083:
2071:
2070:
2023:
2021:
2020:
2015:
1984:
1982:
1981:
1976:
1962:
1961:
1948:
1947:
1946:
1916:
1914:
1913:
1908:
1903:
1902:
1901:
1888:
1887:
1878:
1877:
1867:
1860:
1859:
1839:
1837:
1836:
1824:
1819:
1818:
1809:
1808:
1798:
1791:
1790:
1770:
1769:
1754:
1753:
1743:
1736:
1735:
1696:
1685:
1664:
1662:
1661:
1656:
1651:
1649:
1648:
1636:
1631:
1630:
1615:
1614:
1522:
1473:
1433:
1394:
1384:
1351:
1331:
1326:
1321:
1316:
1311:
1305:
1265:
1231:
1207:
1206:
1180:
1178:
1177:
1172:
1170:
1149:
1147:
1146:
1141:
1139:
1135:
1134:
1109:
1105:
1104:
1075:
1073:
1072:
1067:
1055:
1053:
1052:
1047:
1045:
1044:
1040:
1039:
1020:
1019:
1008:
1003:
985:
981:
980:
943:
941:
940:
935:
896:
895:
879:
877:
876:
871:
869:
868:
852:
850:
849:
844:
839:
838:
820:
819:
807:
806:
791:
762:
760:
759:
754:
715:
711:
710:
688:
687:
671:
669:
668:
663:
658:
657:
639:
638:
626:
625:
598:
596:
595:
590:
578:
576:
575:
570:
565:
564:
546:
545:
533:
532:
448:
441:
437:
434:
428:
426:
392:"Huffman coding"
385:
361:
353:
296:David A. Huffman
268:entropy encoding
245:David A. Huffman
221:computer science
26:
25:
5219:
5218:
5214:
5213:
5212:
5210:
5209:
5208:
5179:
5178:
5177:
5172:
5139:
5123:
5107:
5088:Rateâdistortion
5021:
4950:
4869:
4796:
4717:
4622:
4618:Sub-band coding
4526:
4451:Predictive type
4446:
4371:
4338:LZSS + Huffman
4288:LZ77 + Huffman
4277:
4187:
4123:Dictionary type
4117:
4019:Adaptive coding
3996:
3990:
3943:
3904:
3899:
3898:
3883:
3879:
3864:10.1137/0121057
3841:
3837:
3828:
3826:
3799:
3793:
3789:
3757:
3753:
3738:
3719:10.1.1.589.4726
3698:
3694:
3669:
3665:
3648:
3644:
3627:
3623:
3612:
3608:
3601:
3585:
3581:
3572:
3570:
3562:
3553:
3549:
3523:
3514:
3510:
3505:
3457:
3406:
3403:
3402:
3352:
3349:
3348:
3332:
3326:
3269:
3266:
3265:
3255:problem, after
3220:
3216:
3198:
3194:
3189:
3186:
3185:
3153:
3149:
3131:
3127:
3122:
3119:
3118:
3086:
3082:
3064:
3060:
3055:
3052:
3051:
3019:
3015:
3007:
3004:
3003:
2999:
2991:Richard M. Karp
2968:
2928:
2925:
2924:
2899:
2896:
2895:
2878:
2875:
2874:
2846:
2843:
2842:
2828:
2797:
2793:
2789:
2769:
2760:
2756:
2755:
2751:
2745:
2739:
2736:
2735:
2724:totally ordered
2720:
2701:
2654:
2651:
2650:
2632:
2621:polynomial time
2616:
2541:
2535:
2527:frequency table
2522:
2520:Main properties
2512:
2495:
2491:
2483:
2480:
2479:
2471:
2307:
2304:
2303:
2302:leaf nodes and
2287:
2284:
2283:
2276:two child nodes
2235:
2232:
2231:
2129:
2126:
2125:
2105:
2101:
2092:
2088:
2079:
2075:
2066:
2062:
2057:
2054:
2053:
2042:
2035:
2030:
2028:Basic technique
2000:
1997:
1996:
1957:
1953:
1942:
1938:
1931:
1925:
1922:
1921:
1897:
1893:
1892:
1883:
1879:
1873:
1869:
1855:
1851:
1850:
1832:
1828:
1823:
1814:
1810:
1804:
1800:
1786:
1782:
1781:
1765:
1761:
1749:
1745:
1731:
1727:
1726:
1705:
1702:
1701:
1695:
1687:
1684:
1676:
1644:
1640:
1635:
1626:
1622:
1610:
1606:
1598:
1595:
1594:
1589:
1521:
1513:
1509:
1500:
1498:
1472:
1464:
1460:
1458:
1431:
1429:
1393:
1385:
1383:
1375:
1373:
1350:
1342:
1340:
1329:
1324:
1319:
1314:
1309:
1304:
1296:
1264:
1256:
1230:
1222:
1199:Shannon entropy
1187:
1160:
1155:
1152:
1151:
1124:
1120:
1116:
1094:
1090:
1086:
1081:
1078:
1077:
1061:
1058:
1057:
1035:
1031:
1027:
1015:
1011:
1010:
1004:
993:
970:
966:
962:
957:
954:
953:
951:
946:
945:
891:
887:
885:
882:
881:
864:
860:
858:
855:
854:
834:
830:
815:
811:
802:
798:
781:
776:
773:
772:
770:
765:
764:
706:
702:
698:
683:
679:
677:
674:
673:
653:
649:
634:
630:
621:
617:
606:
603:
602:
600:
584:
581:
580:
560:
556:
541:
537:
528:
524:
513:
510:
509:
507:
501:
466:
449:
438:
432:
429:
386:
384:
374:
362:
351:
339:
292:
247:while he was a
17:
12:
11:
5:
5217:
5207:
5206:
5201:
5196:
5191:
5174:
5173:
5171:
5170:
5155:
5144:
5141:
5140:
5138:
5137:
5131:
5129:
5125:
5124:
5122:
5121:
5115:
5113:
5109:
5108:
5106:
5105:
5100:
5095:
5090:
5085:
5080:
5075:
5070:
5069:
5068:
5058:
5053:
5052:
5051:
5046:
5035:
5033:
5027:
5026:
5023:
5022:
5020:
5019:
5018:
5017:
5012:
5002:
5001:
5000:
4995:
4990:
4982:
4977:
4972:
4967:
4961:
4959:
4952:
4951:
4949:
4948:
4943:
4938:
4933:
4928:
4923:
4918:
4913:
4912:
4911:
4906:
4901:
4890:
4888:
4881:
4875:
4874:
4871:
4870:
4868:
4867:
4866:
4865:
4860:
4855:
4850:
4840:
4835:
4830:
4825:
4820:
4815:
4810:
4804:
4802:
4798:
4797:
4795:
4794:
4789:
4784:
4779:
4774:
4769:
4764:
4759:
4754:
4749:
4744:
4738:
4736:
4729:
4723:
4722:
4719:
4718:
4716:
4715:
4710:
4705:
4704:
4703:
4698:
4693:
4688:
4683:
4673:
4672:
4671:
4661:
4660:
4659:
4654:
4644:
4639:
4633:
4631:
4624:
4623:
4621:
4620:
4615:
4610:
4605:
4600:
4595:
4590:
4585:
4580:
4575:
4570:
4569:
4568:
4563:
4558:
4547:
4545:
4538:
4532:
4531:
4528:
4527:
4525:
4524:
4522:Psychoacoustic
4519:
4518:
4517:
4512:
4507:
4499:
4498:
4497:
4492:
4487:
4482:
4477:
4467:
4466:
4465:
4454:
4452:
4448:
4447:
4445:
4444:
4443:
4442:
4437:
4432:
4422:
4417:
4412:
4411:
4410:
4405:
4394:
4392:
4390:Transform type
4383:
4377:
4376:
4373:
4372:
4370:
4369:
4368:
4367:
4359:
4358:
4357:
4354:
4346:
4345:
4344:
4336:
4335:
4334:
4326:
4325:
4324:
4316:
4315:
4314:
4306:
4305:
4304:
4299:
4294:
4285:
4283:
4279:
4278:
4276:
4275:
4270:
4265:
4260:
4255:
4250:
4249:
4248:
4243:
4233:
4228:
4223:
4222:
4221:
4211:
4206:
4201:
4195:
4193:
4189:
4188:
4186:
4185:
4184:
4183:
4178:
4173:
4168:
4163:
4158:
4153:
4148:
4143:
4133:
4127:
4125:
4119:
4118:
4116:
4115:
4114:
4113:
4108:
4103:
4098:
4088:
4083:
4078:
4073:
4068:
4063:
4058:
4057:
4056:
4051:
4046:
4036:
4031:
4026:
4021:
4015:
4013:
4004:
3998:
3997:
3989:
3988:
3981:
3974:
3966:
3960:
3959:
3954:
3949:
3942:
3941:External links
3939:
3938:
3937:
3920:Clifford Stein
3903:
3900:
3897:
3896:
3877:
3835:
3787:
3762:(1961-01-31).
3751:
3736:
3692:
3681:(2): 228â230.
3663:
3642:
3621:
3606:
3599:
3579:
3547:
3507:
3506:
3504:
3501:
3456:
3453:
3440:
3437:
3434:
3431:
3428:
3425:
3422:
3419:
3416:
3413:
3410:
3386:
3383:
3380:
3377:
3374:
3371:
3368:
3365:
3362:
3359:
3356:
3328:Main article:
3325:
3322:
3310:Adriano Garsia
3291:
3288:
3285:
3282:
3279:
3276:
3273:
3239:
3235:
3232:
3229:
3226:
3223:
3219:
3215:
3211:
3207:
3204:
3201:
3197:
3193:
3172:
3168:
3165:
3162:
3159:
3156:
3152:
3148:
3144:
3140:
3137:
3134:
3130:
3126:
3105:
3101:
3098:
3095:
3092:
3089:
3085:
3081:
3077:
3073:
3070:
3067:
3063:
3059:
3038:
3034:
3031:
3028:
3025:
3022:
3018:
3014:
3011:
2998:
2995:
2967:
2964:
2950:
2947:
2944:
2941:
2938:
2935:
2932:
2912:
2909:
2906:
2903:
2882:
2862:
2859:
2856:
2853:
2850:
2827:
2824:
2810:
2805:
2800:
2796:
2792:
2787:
2784:
2781:
2778:
2775:
2772:
2768:
2763:
2759:
2754:
2748:
2744:
2719:
2716:
2700:
2697:
2664:
2661:
2658:
2631:
2625:
2615:
2612:
2574:block encoding
2534:
2531:
2521:
2518:
2498:
2494:
2490:
2487:
2470:
2467:
2448:
2447:
2436:
2429:
2428:
2425:
2424:
2423:
2420:
2417:
2411:
2408:
2384:) time, where
2362:
2361:
2358:
2357:
2356:
2353:
2350:
2344:
2337:priority queue
2317:
2314:
2311:
2291:
2239:
2210:
2209:
2206:
2202:
2201:
2198:
2194:
2193:
2190:
2186:
2185:
2182:
2178:
2177:
2174:
2157:
2154:
2151:
2148:
2145:
2142:
2139:
2136:
2133:
2113:
2108:
2104:
2100:
2095:
2091:
2087:
2082:
2078:
2074:
2069:
2065:
2061:
2034:
2031:
2029:
2026:
2013:
2010:
2007:
2004:
1974:
1971:
1968:
1965:
1960:
1956:
1952:
1945:
1941:
1937:
1934:
1930:
1918:
1917:
1906:
1900:
1896:
1891:
1886:
1882:
1876:
1872:
1866:
1863:
1858:
1854:
1849:
1845:
1842:
1835:
1831:
1827:
1822:
1817:
1813:
1807:
1803:
1797:
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1780:
1776:
1773:
1768:
1764:
1760:
1757:
1752:
1748:
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1734:
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1715:
1712:
1709:
1691:
1680:
1666:
1665:
1654:
1647:
1643:
1639:
1634:
1629:
1625:
1621:
1618:
1613:
1609:
1605:
1602:
1587:
1577:Shannon (1948)
1575:As defined by
1550:
1549:
1539:
1536:
1533:
1530:
1527:
1524:
1517:
1511:
1505:
1494:
1493:
1490:
1487:
1484:
1481:
1478:
1475:
1468:
1462:
1454:
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1450:
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1441:
1438:
1435:
1426:
1422:
1421:
1411:
1408:
1405:
1402:
1399:
1396:
1389:
1379:
1369:
1368:
1365:
1362:
1359:
1356:
1353:
1346:
1336:
1335:
1332:
1327:
1322:
1317:
1312:
1307:
1300:
1293:
1286:
1285:
1282:
1279:
1276:
1273:
1270:
1267:
1260:
1252:
1251:
1248:
1245:
1242:
1239:
1236:
1233:
1226:
1219:
1186:
1183:
1169:
1166:
1163:
1159:
1138:
1133:
1130:
1127:
1123:
1119:
1115:
1112:
1108:
1103:
1100:
1097:
1093:
1089:
1085:
1065:
1043:
1038:
1034:
1030:
1026:
1023:
1018:
1014:
1007:
1002:
999:
996:
992:
988:
984:
979:
976:
973:
969:
965:
961:
933:
930:
927:
924:
921:
918:
915:
912:
909:
906:
903:
899:
894:
890:
867:
863:
842:
837:
833:
829:
826:
823:
818:
814:
810:
805:
801:
797:
794:
790:
787:
784:
780:
752:
749:
746:
743:
740:
737:
734:
731:
728:
725:
722:
718:
714:
709:
705:
701:
697:
694:
691:
686:
682:
661:
656:
652:
648:
645:
642:
637:
633:
629:
624:
620:
616:
613:
610:
588:
568:
563:
559:
555:
552:
549:
544:
540:
536:
531:
527:
523:
520:
517:
500:
497:
496:
495:
480:
477:
470:
465:
462:
451:
450:
365:
363:
356:
350:
347:
338:
335:
327:Claude Shannon
311:Robert M. Fano
291:
288:
241:Huffman coding
213:
212:
209:
206:
202:
201:
198:
195:
191:
190:
187:
184:
180:
179:
176:
173:
169:
168:
165:
162:
158:
157:
154:
151:
147:
146:
143:
140:
136:
135:
132:
129:
125:
124:
121:
118:
114:
113:
110:
107:
103:
102:
99:
96:
92:
91:
88:
85:
81:
80:
77:
74:
70:
69:
66:
63:
59:
58:
55:
52:
48:
47:
44:
41:
37:
36:
33:
30:
15:
9:
6:
4:
3:
2:
5216:
5205:
5202:
5200:
5197:
5195:
5192:
5190:
5187:
5186:
5184:
5168:
5164:
5156:
5154:
5146:
5145:
5142:
5136:
5133:
5132:
5130:
5126:
5120:
5117:
5116:
5114:
5110:
5104:
5101:
5099:
5096:
5094:
5091:
5089:
5086:
5084:
5081:
5079:
5076:
5074:
5071:
5067:
5064:
5063:
5062:
5059:
5057:
5054:
5050:
5047:
5045:
5042:
5041:
5040:
5037:
5036:
5034:
5032:
5028:
5016:
5013:
5011:
5008:
5007:
5006:
5003:
4999:
4996:
4994:
4991:
4989:
4986:
4985:
4983:
4981:
4978:
4976:
4973:
4971:
4968:
4966:
4963:
4962:
4960:
4957:
4953:
4947:
4946:Video quality
4944:
4942:
4939:
4937:
4934:
4932:
4929:
4927:
4924:
4922:
4919:
4917:
4914:
4910:
4907:
4905:
4902:
4900:
4897:
4896:
4895:
4892:
4891:
4889:
4885:
4882:
4880:
4876:
4864:
4861:
4859:
4856:
4854:
4851:
4849:
4846:
4845:
4844:
4841:
4839:
4836:
4834:
4831:
4829:
4826:
4824:
4821:
4819:
4816:
4814:
4811:
4809:
4806:
4805:
4803:
4799:
4793:
4790:
4788:
4785:
4783:
4780:
4778:
4775:
4773:
4770:
4768:
4765:
4763:
4760:
4758:
4755:
4753:
4750:
4748:
4745:
4743:
4740:
4739:
4737:
4733:
4730:
4728:
4724:
4714:
4711:
4709:
4706:
4702:
4699:
4697:
4694:
4692:
4689:
4687:
4684:
4682:
4679:
4678:
4677:
4674:
4670:
4667:
4666:
4665:
4662:
4658:
4655:
4653:
4650:
4649:
4648:
4645:
4643:
4640:
4638:
4635:
4634:
4632:
4629:
4625:
4619:
4616:
4614:
4613:Speech coding
4611:
4609:
4608:Sound quality
4606:
4604:
4601:
4599:
4596:
4594:
4591:
4589:
4586:
4584:
4583:Dynamic range
4581:
4579:
4576:
4574:
4571:
4567:
4564:
4562:
4559:
4557:
4554:
4553:
4552:
4549:
4548:
4546:
4542:
4539:
4537:
4533:
4523:
4520:
4516:
4513:
4511:
4508:
4506:
4503:
4502:
4500:
4496:
4493:
4491:
4488:
4486:
4483:
4481:
4478:
4476:
4473:
4472:
4471:
4468:
4464:
4461:
4460:
4459:
4456:
4455:
4453:
4449:
4441:
4438:
4436:
4433:
4431:
4428:
4427:
4426:
4423:
4421:
4418:
4416:
4413:
4409:
4406:
4404:
4401:
4400:
4399:
4396:
4395:
4393:
4391:
4387:
4384:
4382:
4378:
4366:
4363:
4362:
4360:
4355:
4353:
4350:
4349:
4348:LZ77 + Range
4347:
4343:
4340:
4339:
4337:
4333:
4330:
4329:
4327:
4323:
4320:
4319:
4317:
4313:
4310:
4309:
4307:
4303:
4300:
4298:
4295:
4293:
4290:
4289:
4287:
4286:
4284:
4280:
4274:
4271:
4269:
4266:
4264:
4261:
4259:
4256:
4254:
4251:
4247:
4244:
4242:
4239:
4238:
4237:
4234:
4232:
4229:
4227:
4224:
4220:
4217:
4216:
4215:
4212:
4210:
4207:
4205:
4202:
4200:
4197:
4196:
4194:
4190:
4182:
4179:
4177:
4174:
4172:
4169:
4167:
4164:
4162:
4159:
4157:
4154:
4152:
4149:
4147:
4144:
4142:
4139:
4138:
4137:
4134:
4132:
4129:
4128:
4126:
4124:
4120:
4112:
4109:
4107:
4104:
4102:
4099:
4097:
4094:
4093:
4092:
4089:
4087:
4084:
4082:
4079:
4077:
4074:
4072:
4069:
4067:
4064:
4062:
4059:
4055:
4052:
4050:
4047:
4045:
4042:
4041:
4040:
4037:
4035:
4032:
4030:
4027:
4025:
4022:
4020:
4017:
4016:
4014:
4012:
4008:
4005:
4003:
3999:
3994:
3987:
3982:
3980:
3975:
3973:
3968:
3967:
3964:
3958:
3955:
3953:
3950:
3948:
3945:
3944:
3935:
3934:0-262-03293-7
3931:
3927:
3926:
3921:
3917:
3913:
3909:
3906:
3905:
3891:
3887:
3881:
3873:
3869:
3865:
3861:
3857:
3853:
3849:
3848:Tucker, A. C.
3845:
3839:
3825:
3821:
3817:
3813:
3809:
3805:
3798:
3791:
3783:
3779:
3775:
3771:
3766:
3761:
3755:
3747:
3743:
3739:
3737:0-8186-8132-2
3733:
3729:
3725:
3720:
3715:
3711:
3707:
3703:
3696:
3688:
3684:
3680:
3676:
3675:
3667:
3658:
3653:
3646:
3638:
3637:
3632:
3625:
3618:
3617:
3610:
3602:
3596:
3592:
3591:
3583:
3568:
3561:
3557:
3551:
3543:
3539:
3535:
3531:
3530:
3522:
3518:
3512:
3508:
3500:
3498:
3494:
3490:
3486:
3482:
3478:
3474:
3469:
3466:
3461:
3452:
3435:
3432:
3429:
3426:
3423:
3420:
3417:
3414:
3411:
3400:
3381:
3378:
3375:
3372:
3369:
3366:
3363:
3360:
3357:
3346:
3342:
3338:
3331:
3321:
3319:
3315:
3311:
3307:
3303:
3286:
3283:
3280:
3277:
3271:
3262:
3258:
3254:
3237:
3233:
3230:
3227:
3224:
3221:
3217:
3213:
3209:
3205:
3202:
3199:
3195:
3191:
3170:
3166:
3163:
3160:
3157:
3154:
3150:
3146:
3142:
3138:
3135:
3132:
3128:
3124:
3103:
3099:
3096:
3093:
3090:
3087:
3083:
3079:
3075:
3071:
3068:
3065:
3061:
3057:
3036:
3032:
3029:
3026:
3023:
3020:
3016:
3012:
3009:
2994:
2992:
2988:
2983:
2979:
2977:
2973:
2963:
2961:
2945:
2942:
2939:
2936:
2930:
2907:
2901:
2880:
2857:
2854:
2848:
2840:
2836:
2832:
2823:
2808:
2803:
2798:
2794:
2790:
2766:
2761:
2757:
2752:
2746:
2732:
2728:
2725:
2715:
2713:
2708:
2707:
2696:
2694:
2690:
2686:
2682:
2678:
2662:
2659:
2656:
2648:
2644:
2640:
2638:
2629:
2624:
2622:
2611:
2609:
2605:
2604:Golomb coding
2601:
2597:
2592:
2588:
2586:
2581:
2579:
2575:
2571:
2566:
2563:
2557:
2555:
2551:
2547:
2540:
2530:
2528:
2517:
2496:
2492:
2488:
2485:
2477:
2469:Decompression
2466:
2462:
2460:
2456:
2451:
2445:
2441:
2437:
2434:
2433:
2432:
2426:
2421:
2418:
2415:
2414:
2412:
2409:
2406:
2405:
2404:
2402:
2398:
2394:
2389:
2387:
2383:
2379:
2375:
2371:
2367:
2359:
2354:
2351:
2348:
2347:
2345:
2342:
2341:
2340:
2338:
2333:
2329:
2315:
2312:
2309:
2289:
2281:
2277:
2273:
2269:
2265:
2261:
2257:
2256:internal node
2253:
2237:
2229:
2225:
2216:
2207:
2204:
2203:
2199:
2196:
2195:
2191:
2188:
2187:
2183:
2180:
2179:
2175:
2172:
2171:
2152:
2149:
2146:
2143:
2140:
2137:
2134:
2106:
2102:
2098:
2093:
2089:
2085:
2080:
2076:
2072:
2067:
2063:
2050:
2039:
2025:
2008:
2002:
1993:
1991:
1986:
1972:
1969:
1966:
1963:
1958:
1954:
1950:
1943:
1939:
1932:
1904:
1898:
1894:
1889:
1884:
1880:
1874:
1870:
1864:
1861:
1856:
1852:
1847:
1843:
1840:
1833:
1829:
1825:
1820:
1815:
1811:
1805:
1801:
1795:
1792:
1787:
1783:
1778:
1774:
1766:
1762:
1755:
1750:
1746:
1740:
1737:
1732:
1728:
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1719:
1713:
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1698:
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1690:
1683:
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1652:
1645:
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1637:
1632:
1627:
1623:
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1611:
1607:
1600:
1593:
1592:
1591:
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1578:
1573:
1571:
1567:
1563:
1562:
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1547:
1543:
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1516:
1508:
1504:
1496:
1495:
1471:
1467:
1456:
1455:
1427:
1423:
1419:
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1388:
1382:
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1371:
1370:
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1229:
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1213:
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1200:
1196:
1192:
1182:
1167:
1164:
1161:
1157:
1150:for any code
1136:
1131:
1128:
1125:
1121:
1117:
1113:
1110:
1106:
1101:
1098:
1095:
1091:
1087:
1083:
1076:. Condition:
1063:
1041:
1036:
1032:
1028:
1024:
1021:
1016:
1012:
1005:
1000:
997:
994:
990:
986:
982:
977:
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971:
967:
963:
959:
949:
928:
925:
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916:
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892:
888:
865:
861:
835:
831:
827:
824:
821:
816:
812:
808:
803:
799:
792:
788:
785:
782:
778:
768:
747:
744:
741:
738:
735:
732:
729:
723:
720:
716:
712:
707:
703:
699:
695:
692:
689:
684:
680:
654:
650:
646:
643:
640:
635:
631:
627:
622:
618:
611:
608:
586:
561:
557:
553:
550:
547:
542:
538:
534:
529:
525:
518:
515:
505:
493:
489:
485:
481:
478:
475:
471:
468:
467:
457:
447:
444:
436:
433:December 2021
425:
422:
418:
415:
411:
408:
404:
401:
397:
394: â
393:
389:
388:Find sources:
382:
378:
372:
371:
366:This article
364:
360:
355:
354:
346:
344:
334:
332:
328:
323:
321:
316:
313:, assigned a
312:
308:
304:
301:
297:
287:
285:
281:
277:
273:
269:
265:
261:
256:
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250:
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210:
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78:
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72:
71:
67:
64:
61:
60:
56:
53:
50:
49:
45:
42:
39:
38:
34:
31:
28:
27:
21:
5204:Binary trees
5119:Hutter Prize
5083:Quantization
4988:Compensation
4782:Quantization
4505:Compensation
4071:ShannonâFano
4038:
4011:Entropy type
3923:
3902:Bibliography
3889:
3880:
3855:
3851:
3838:
3827:. Retrieved
3807:
3803:
3790:
3773:
3769:
3754:
3701:
3695:
3678:
3672:
3666:
3645:
3634:
3624:
3615:
3609:
3589:
3582:
3571:. Retrieved
3566:
3550:
3533:
3527:
3511:
3497:quantization
3470:
3464:
3458:
3455:Applications
3398:
3340:
3336:
3333:
3252:
3000:
2981:
2980:
2975:
2971:
2969:
2830:
2829:
2730:
2721:
2704:
2702:
2692:
2688:
2684:
2680:
2676:
2646:
2642:
2639:-ary Huffman
2636:
2635:
2633:
2627:
2617:
2593:
2589:
2582:
2570:power of two
2567:
2558:
2542:
2523:
2472:
2463:
2458:
2454:
2452:
2449:
2443:
2439:
2430:
2396:
2390:
2385:
2381:
2377:
2373:
2372:leaves has 2
2369:
2365:
2363:
2334:
2330:
2279:
2275:
2271:
2267:
2263:
2262:itself, the
2259:
2221:
1994:
1987:
1919:
1692:
1688:
1681:
1677:
1672:
1667:
1584:
1580:
1574:
1569:
1565:
1560:
1555:
1553:
1545:
1541:
1518:
1514:
1506:
1502:
1469:
1465:
1417:
1413:
1390:
1386:
1380:
1376:
1347:
1343:
1301:
1297:
1290:
1261:
1257:
1227:
1223:
1215:
1211:
1201:
1194:
1190:
1188:
947:
766:
503:
502:
474:proportional
439:
430:
420:
413:
406:
399:
387:
375:Please help
370:verification
367:
340:
324:
293:
263:
257:
240:
229:Huffman code
228:
218:
5078:Prefix code
4931:Frame types
4752:Color space
4578:Convolution
4308:LZ77 + ANS
4219:Incremental
4192:Other types
4111:Levenshtein
3517:Huffman, D.
3261:Alan Tucker
2393:linear-time
2274:, links to
2224:binary tree
2033:Compression
1425:Optimality
1295:Codewords (
343:prefix code
337:Terminology
320:binary tree
251:student at
233:prefix code
5183:Categories
5135:Mark Adler
5093:Redundancy
5010:Daubechies
4993:Estimation
4926:Frame rate
4848:Daubechies
4808:Chain code
4767:Macroblock
4573:Companding
4510:Estimation
4430:Daubechies
4136:LempelâZiv
4096:Exp-Golomb
4024:Arithmetic
3858:(4): 514.
3829:2024-09-10
3657:1604.07476
3573:2014-02-20
3503:References
2987:Morse code
2614:Variations
2537:See also:
2533:Optimality
1548:) = 2.205
403:newspapers
315:term paper
5112:Community
4936:Interlace
4322:Zstandard
4101:Fibonacci
4091:Universal
4049:Canonical
3844:Hu, T. C.
3746:124587565
3714:CiteSeerX
3569:: 382â410
3284:
3253:HuâTucker
2943:
2489:⋅
2313:−
2252:leaf node
1964:
1936:→
1890:
1848:∑
1844:−
1821:
1779:∑
1724:∑
1633:
1420:) = 2.25
1255:Weights (
1111:≤
1025:
991:∑
923:…
905:∈
825:…
742:…
724:∈
696:
644:…
551:…
508:Alphabet
294:In 1951,
5098:Symmetry
5066:Timeline
5049:FM-index
4894:Bit rate
4887:Concepts
4735:Concepts
4598:Sampling
4551:Bit rate
4544:Concepts
4246:Sequitur
4081:Tunstall
4054:Modified
4044:Adaptive
4002:Lossless
3639:: 54â58.
3558:(1976).
3519:(1952).
3487:such as
3257:T. C. Hu
2873:, where
2576:, e.g.,
2044:message.
1570:biunique
1566:complete
1556:biunique
1221:Symbol (
488:expected
298:and his
5056:Entropy
5005:Wavelet
4984:Motion
4843:Wavelet
4823:Fractal
4818:Deflate
4801:Methods
4588:Latency
4501:Motion
4425:Wavelet
4342:LHA/LZH
4292:Deflate
4241:Re-Pair
4236:Grammar
4066:Shannon
4039:Huffman
3995:methods
3872:2099603
3824:2265146
3477:Deflate
2215:entropy
1670:entropy
1492:
1452:= 1.00
1334:
1289:Output
1210:Input (
1185:Example
417:scholar
290:History
5167:codecs
5128:People
5031:Theory
4998:Vector
4515:Vector
4332:Brotli
4282:Hybrid
4181:Snappy
4034:Golomb
3932:
3918:, and
3870:
3822:
3744:
3734:
3716:
3597:
3485:codecs
2839:greedy
2585:dyadic
2562:patent
2401:queues
2280:parent
2272:weight
2268:parent
2264:weight
2260:symbol
2254:or an
2173:Symbol
1538:0.518
1535:0.423
1532:0.521
1529:0.411
1526:0.332
1022:length
767:Output
693:weight
601:Tuple
419:
412:
405:
398:
390:
272:linear
264:weight
211:10010
200:00111
189:11000
178:10011
167:00110
156:11001
4958:parts
4956:Codec
4921:Frame
4879:Video
4863:SPIHT
4772:Pixel
4727:Image
4681:ACELP
4652:ADPCM
4642:Îź-law
4637:A-law
4630:parts
4628:Codec
4536:Audio
4475:ACELP
4463:ADPCM
4440:SPIHT
4381:Lossy
4365:bzip2
4356:LZHAM
4312:LZFSE
4214:Delta
4106:Gamma
4086:Unary
4061:Range
3868:JSTOR
3820:S2CID
3800:(PDF)
3742:S2CID
3652:arXiv
3567:ICALP
3563:(PDF)
3524:(PDF)
3481:PKZIP
3302:-time
2578:ASCII
2228:array
2176:Code
1489:1.79
1486:2.64
1483:1.74
1480:2.74
1477:3.32
1410:0.58
1407:0.32
1404:0.60
1401:0.45
1398:0.30
1281:0.29
1278:0.16
1275:0.30
1272:0.15
1269:0.10
771:Code
504:Input
469:Given
424:JSTOR
410:books
249:Sc.D.
145:0110
134:1011
123:0010
112:0111
101:1000
90:1010
79:1101
40:space
35:Code
4970:DPCM
4777:PSNR
4708:MDCT
4701:WLPC
4686:CELP
4647:DPCM
4495:WLPC
4480:CELP
4458:DPCM
4408:MDCT
4352:LZMA
4253:LDCT
4231:DPCM
4176:LZWL
4166:LZSS
4161:LZRW
4151:LZJB
3930:ISBN
3732:ISBN
3710:IEEE
3595:ISBN
3491:and
3489:JPEG
3312:and
3259:and
2634:The
2380:log
2218:two.
2208:111
2200:110
2153:0.05
2141:0.35
1862:>
1793:>
1738:>
1668:The
1474:) â
1461:âlog
1449:1/4
1446:1/4
1443:1/4
1440:1/8
1437:1/8
1284:= 1
1250:Sum
952:Let
948:Goal
479:Find
396:news
307:exam
227:, a
223:and
68:000
57:010
46:111
32:Freq
29:Char
5015:DWT
4965:DCT
4909:VBR
4904:CBR
4899:ABR
4858:EZW
4853:DWT
4838:RLE
4828:KLT
4813:DCT
4696:LSP
4691:LAR
4676:LPC
4669:FFT
4566:VBR
4561:CBR
4556:ABR
4490:LSP
4485:LAR
4470:LPC
4435:DWT
4420:FFT
4415:DST
4403:DCT
4302:LZS
4297:LZX
4273:RLE
4268:PPM
4263:PAQ
4258:MTF
4226:DMC
4204:CTW
4199:BWT
4171:LZW
4156:LZO
4146:LZ4
4141:842
3860:doi
3812:doi
3778:doi
3724:doi
3683:doi
3538:doi
3493:MP3
3418:111
3412:110
3364:001
3358:000
3308:of
3281:log
3184:or
2940:log
2923:or
2743:max
2395:(O(
2192:10
2147:0.2
2135:0.4
1955:log
1929:lim
1881:log
1812:log
1624:log
1510:log
1315:011
1310:010
379:by
300:MIT
282:or
253:MIT
219:In
5185::
4833:LP
4664:FT
4657:DM
4209:CM
3922:.
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2657:n
2647:n
2643:n
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