2527:
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).
2485:
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
2049:
5170:
5160:
2003:, 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.
2060:
3000:, 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
2179:. 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:
370:
31:
467:
356:(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.
2602:
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.
2598:. 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.
3345:
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
2720:
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
2601:
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
2570:
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
2526:
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
34:
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
2629:
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
2342:
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
2484:
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
2475:
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
2054:
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
2575:
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.
3478:
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.
1926:
2414:, 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:
2571:
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
2554:
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
2228:
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
1065:
2744:
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
772:
1714:
328:
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
2343:
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.
1159:
3012:
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,
2476:
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.
2989:, 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.
2995:
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
2535:
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
862:
3260:
3193:
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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.
2468:
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
2733:
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
1674:
2844:
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
1994:
953:
681:
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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.
3807:
2981:
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
3059:
2177:
3460:
3406:
1575:, 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
1579:
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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1190:
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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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35:
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.
2702:
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
3315:
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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889:
2932:
2033:
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2337:
2006:
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
273:
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 (
3994:
2902:
2609:. 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
2311:
2259:
1085:
608:
686:
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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
2617:
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
4662:
4473:
2540:
must be stored with the compressed text. See the Decompression section above for more information about the various techniques employed for this purpose.
2035:
for that probability distribution. (However, for each minimizing codeword length assignment, there exists at least one Huffman code with those lengths.)
4362:
2430:
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.
4176:
2293:
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
1090:
5209:
4868:
4691:
4485:
3684:
2560:
3327:(1977), uses simpler logic to perform the same comparisons in the same total time bound. These optimal alphabetic binary trees are often used as
285:
to the number of input weights if these weights are sorted. However, although optimal among methods encoding symbols separately, Huffman coding
4873:
4450:
1587:
1921:{\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}}.}
3609:
2461:
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.
263:
4603:
2421:
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).
1200:
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
2698:-ary tree for Huffman coding. In these cases, additional 0-probability place holders must be added. This is because the tree must form an
4980:
4718:
4657:
4468:
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4241:
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Once the Huffman tree has been generated, it is traversed to generate a dictionary which maps the symbols to binary codes as follows:
2000:
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3987:
3944:
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453:
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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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3507:
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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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413:
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3316:
1934:
2660:-ary tree. This approach was considered by Huffman in his original paper. The same algorithm applies as for binary (
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5025:
4863:
4843:
4787:
4445:
4236:
4039:
3935:
894:
294:
3682:
Gallager, R.G.; van Voorhis, D.C. (1975). "Optimal source codes for geometrically distributed integer alphabets".
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4263:
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247:
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Abrahams, J. (1997-06-11). "Code and parse trees for lossless source encoding". Written at Arlington, VA, USA.
420:
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341:
2052:
Visualisation of the use of Huffman coding to encode the message "A_DEAD_DAD_CEDED_A_BAD_BABE_A_BEADED_ABACA_
5054:
4425:
4268:
4064:
4054:
3016:
2845:
2716:
2621:. However, run-length coding is not as adaptable to as many input types as other compression technologies.
2618:
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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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402:
4679:
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4214:
4059:
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3539:
3415:
3361:
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2486:
2411:
1060:{\textstyle L\left(C\left(W\right)\right)=\sum _{i=1}^{n}{w_{i}\operatorname {length} \left(c_{i}\right)}}
5214:
5083:
3808:"A Dynamic Programming Algorithm for Constructing Optimal Prefix-Free Codes with Unequal Letter Costs"
352:
Huffman coding uses a specific method for choosing the representation for each symbol, resulting in a
4767:
4229:
4191:
4012:
3922:
1570:
1164:
270:
3729:
2748:
683:, 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.
313:
235:
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2937:
2579:
For a set of symbols with a uniform probability distribution and a number of members which is a
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least probable symbols are taken together, instead of just the 2 least probable. Note that for
2048:
266:, and published in the 1952 paper "A Method for the Construction of Minimum-Redundancy Codes".
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3734:
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3661:
Gribov, Alexander (2017-04-10). "Optimal Compression of a Polyline with Segments and Arcs".
3641:
3548:
3527:
3408:, which, having the same codeword lengths as the original solution, is also optimal. But in
2849:
2464:
In many cases, time complexity is not very important in the choice of algorithm here, since
2224:
The standard way to represent a signal made of 4 symbols is by using 2 bits/symbol, but the
427:
306:
278:
255:
231:
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1996:. So for simplicity, symbols with zero probability can be left out of the formula above.)
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Prefix codes nevertheless remain in wide use because of their simplicity, high speed, and
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2908:
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2009:
1209:
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767:{\displaystyle w_{i}=\operatorname {weight} \left(a_{i}\right),\,i\in \{1,2,\dots ,n\}}
593:
498:
337:
321:
3354:, since it is optimal like Huffman coding, but alphabetic in weight probability, like
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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
2427:
Dequeue the two nodes with the lowest weight by examining the fronts of both queues.
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4096:
4071:
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3834:
3627:
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
1931:(Note: A symbol with zero probability has zero contribution to the entropy, since
3599:
5088:
4966:
4762:
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3861:(1971). "Optimal Computer Search Trees and Variable-Length Alphabetical Codes".
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1154:{\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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30:
17:
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2563:, a single code may be insufficient for optimality. Other methods such as
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3899:(1998), "Algorithm G (GarsiaâWachs algorithm for optimum binary trees)",
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3267:
3957:
2706:−1, then the set of source words will form a proper Huffman tree.
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4302:
4251:
3882:
3642:"Profile: David A. Huffman: Encoding the "Neatness" of Ones and Zeroes"
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3826:
3486:. They are often used as a "back-end" to other compression methods.
4342:
2737:
2572:
2438:
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.
2277:(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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465:
29:
2489:, the compression model can be precisely reconstructed with just
4332:
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4171:
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3715:. Division of Mathematics, Computer & Information Sciences,
3499:
3358:. The HuffmanâShannonâFano code corresponding to the example is
2457:. Repeat the process at both the left child and the right child.
2449:
If node is not a leaf node, label the edge to the left child as
317:
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4307:
4273:
3901:
The Art of Computer Programming, Vol. 3: Sorting and Searching
3776:"Minimum-redundancy coding for the discrete noiseless channel"
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363:
310:
2837:
Length-limited Huffman coding/minimum variance Huffman coding
2375:
Since efficient priority queue data structures require O(log
2371:
The remaining node is the root node and the tree is complete.
277:) for each possible value of the source symbol. As in other
316:
classmates were given the choice of a term paper or a final
3532:"A Method for the Construction of Minimum-Redundancy Codes"
1215:
of the given set of weights; the result is nearly optimal.
857:{\displaystyle C\left(W\right)=(c_{1},c_{2},\dots ,c_{n})}
2269:. Initially, all nodes are leaf nodes, which contain the
3255:{\displaystyle H\left(A,C\right)=\left\{0,10,11\right\}}
3188:{\displaystyle H\left(A,C\right)=\left\{00,01,1\right\}}
3121:{\displaystyle H\left(A,C\right)=\left\{00,1,01\right\}}
3598:
Ze-Nian Li; Mark S. Drew; Jiangchuan Liu (2014-04-09).
2433:
Enqueue the new node into the rear of the second queue.
269:
The output from Huffman's algorithm can be viewed as a
2740:, meaning a way to order weights and to add them. The
2241:, the size of which depends on the number of symbols,
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336:
In doing so, Huffman outdid Fano, who had worked with
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2725:, 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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2012:
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among all compression methods - it is replaced with
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Huffman coding in various languages on Rosetta Code
3939:, Second Edition. MIT Press and McGraw-Hill, 2001.
501:codeword length (equivalently, a tree with minimum
394:. Unsourced material may be challenged and removed.
333:and quickly proved this method the most efficient.
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3008:Optimal alphabetic binary trees (HuâTucker coding)
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1686:(in bits) is the weighted sum, across all symbols
1669:{\displaystyle h(a_{i})=\log _{2}{1 \over w_{i}}.}
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1059:
947:
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864:, which is the tuple of (binary) codewords, where
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675:
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3905:. See also History and bibliography, pp. 453â454.
2424:While there is more than one node in the queues:
250:. The process of finding or using such a code is
3903:(2nd ed.), AddisonâWesley, pp. 451â453
3274:, the authors of the paper presenting the first
2753:
2583:, Huffman coding is equivalent to simple binary
2357:While there is more than one node in the queue:
1939:
2605:A practical alternative, in widespread use, is
2418:Start with as many leaves as there are symbols.
1989:{\displaystyle \lim _{w\to 0^{+}}w\log _{2}w=0}
3625:J. Duda, K. Tahboub, N. J. Gadil, E. J. Delp,
3988:
2833:, a problem first applied to circuit design.
2656:â 1} alphabet to encode message and build an
2410:)) method to create a Huffman tree using two
1708:, of the information content of each symbol:
948:{\displaystyle a_{i},\,i\in \{1,2,\dots ,n\}}
8:
3449:
3419:
3395:
3365:
2166:
2142:
2122:
2070:
942:
918:
761:
737:
676:{\displaystyle W=(w_{1},w_{2},\dots ,w_{n})}
583:{\displaystyle A=(a_{1},a_{2},\dots ,a_{n})}
483:A set of symbols and their weights (usually
2346:The simplest construction algorithm uses a
2237:of nodes. These can be stored in a regular
2128:{\displaystyle \{a_{1},a_{2},a_{3},a_{4}\}}
297:if a better compression ratio is required.
4894:
4742:
4551:
4396:
4017:
3995:
3981:
3973:
2567:often have better compression capability.
36:
3728:
3666:
3604:. Springer Science & Business Media.
3417:
3363:
3280:
3200:
3133:
3066:
3018:
2939:
2910:
2889:
2857:
2808:
2780:
2771:
2756:
2750:
2665:
2506:
2494:
2318:
2298:
2246:
2140:
2116:
2103:
2090:
2077:
2068:
2011:
1968:
1953:
1942:
1936:
1908:
1903:
1894:
1884:
1866:
1861:
1843:
1834:
1825:
1815:
1797:
1792:
1776:
1760:
1742:
1737:
1716:
1655:
1646:
1637:
1621:
1609:
1166:
1092:
1072:
1046:
1026:
1021:
1015:
1004:
968:
911:
902:
896:
875:
869:
845:
826:
813:
787:
730:
717:
694:
688:
664:
645:
632:
617:
595:
571:
552:
539:
524:
454:Learn how and when to remove this message
3128:, but instead should be assigned either
2993:Huffman coding with unequal letter costs
2977:Huffman coding with unequal letter costs
2446:Start with current node set to the root.
2180:
2047:
1217:
3821:(5) (published 1998-09-01): 1770â1781.
3815:IEEE Transactions on Information Theory
3685:IEEE Transactions on Information Theory
3519:
2561:independent and identically distributed
2559:are unknown. Also, if symbols are not
2550:Arithmetic coding § Huffman coding
2387:â1 nodes, this algorithm operates in O(
2063:A source generates 4 different symbols
590:, which is the symbol alphabet of size
3781:IRE Transactions on Information Theory
3571:"On the construction of Huffman trees"
3054:{\displaystyle A=\left\{a,b,c\right\}}
2379:) time per insertion, and a tree with
3963:Huffman codes (python implementation)
2172:{\displaystyle \{0.4;0.35;0.2;0.05\}}
7:
3455:{\displaystyle \{110,111,00,01,10\}}
3401:{\displaystyle \{000,001,01,10,11\}}
1383:Contribution to weighted path length
1204:over all codes, but we will compute
1067:be the weighted path length of code
392:adding citations to reliable sources
3863:SIAM Journal on Applied Mathematics
3806:Golin, Mordekai J. (January 1998).
2453:and the edge to the right child as
2985:digits will always have a cost of
2848:solves this problem with a simple
2796:
2793:
2790:
2787:
2784:
2781:
2233:The technique works by creating a
503:weighted path length from the root
497:(a set of codewords) with minimum
25:
3968:A visualization of Huffman coding
3947:. Section 16.3, pp. 385â392.
3630:, Picture Coding Symposium, 2015.
5169:
5168:
5159:
5158:
368:
242:is a particular type of optimal
5210:Lossless compression algorithms
2001:Shannon's source coding theorem
1185:{\displaystyle T\left(W\right)}
379:needs additional citations for
286:
3300:
3285:
2959:
2944:
2921:
2915:
2871:
2862:
2826:{\displaystyle \max _{i}\left}
2652:algorithm uses the {0, 1,...,
2366:Add the new node to the queue.
2022:
2016:
1946:
1782:
1769:
1727:
1721:
1627:
1614:
851:
806:
670:
625:
577:
532:
1:
3494:'s algorithm) and multimedia
2842:Length-limited Huffman coding
1601:with non-null probability is
1468:Information content (in bits)
3262:. This is also known as the
2515:{\displaystyle B\cdot 2^{B}}
1551:
1548:
1545:
1542:
1539:
1536:
1502:
1499:
1496:
1493:
1490:
1487:
1462:
1459:
1456:
1453:
1450:
1447:
1423:
1420:
1417:
1414:
1411:
1408:
1377:
1374:
1371:
1368:
1365:
1339:
1334:
1329:
1324:
1319:
1294:
1291:
1288:
1285:
1282:
1279:
1257:
1254:
1251:
1248:
1245:
254:, an algorithm developed by
3506:have a front-end model and
3352:HuffmanâShannonâFano coding
3061:could not be assigned code
2522:bits of information (where
1569:, meaning that the code is
470:Constructing a Huffman Tree
5233:
5050:Compressed data structures
4372:RLE + BWT + MTF + Huffman
4040:Asymmetric numeral systems
3936:Introduction to Algorithms
3739:10.1109/SEQUEN.1997.666911
3601:Fundamentals of Multimedia
3553:10.1109/JRPROC.1952.273898
3338:
3335:The canonical Huffman code
3306:{\displaystyle O(n\log n)}
2965:{\displaystyle O(n\log n)}
2742:Huffman template algorithm
2729:Huffman template algorithm
2723:adaptive arithmetic coding
2557:probability mass functions
2547:
2399:is the number of symbols.
2289:and an optional link to a
1697:with non-zero probability
1590:, the information content
295:asymmetric numeral systems
246:that is commonly used for
27:Technique to compress data
5154:
4409:Discrete cosine transform
4339:LZ77 + Huffman + context
2686:) codes, except that the
2261:. A node can be either a
1594:(in bits) of each symbol
1435:
1350:Codeword length (in bits)
1344:
1299:
1220:
248:lossless data compression
5114:Smallest grammar problem
3793:10.1109/TIT.1961.1057615
3717:Office of Naval Research
3698:10.1109/TIT.1975.1055357
3346:is sometimes called the
2472:grows to be very large.
5055:Compressed suffix array
4604:NyquistâShannon theorem
3484:lack of patent coverage
2846:package-merge algorithm
2717:adaptive Huffman coding
2710:Adaptive Huffman coding
2619:modified Huffman coding
1508:Contribution to entropy
495:prefix-free binary code
3540:Proceedings of the IRE
3456:
3410:canonical Huffman code
3402:
3348:canonical Huffman code
3341:Canonical Huffman code
3317:GarsiaâWachs algorithm
3307:
3256:
3189:
3122:
3055:
2966:
2928:
2898:
2878:
2827:
2680:
2516:
2333:
2307:
2255:
2230:
2173:
2129:
2056:
2029:
1990:
1922:
1670:
1208:and compare it to the
1186:
1155:
1081:
1061:
1020:
949:
885:
858:
768:
677:
604:
584:
510:Formalized description
471:
227:
5084:Kolmogorov complexity
4952:Video characteristics
4329:LZ77 + Huffman + ANS
3640:Huffman, Ken (1991).
3457:
3403:
3308:
3257:
3190:
3123:
3056:
2967:
2929:
2899:
2879:
2877:{\displaystyle O(nL)}
2828:
2681:
2517:
2334:
2308:
2256:
2174:
2130:
2062:
2051:
2030:
1991:
1923:
1671:
1565:For any code that is
1187:
1156:
1082:
1062:
1000:
950:
886:
884:{\displaystyle c_{i}}
859:
769:
678:
605:
585:
469:
287:is not always optimal
33:
5174:Compression software
4768:Compression artifact
4724:Psychoacoustic model
3923:Charles E. Leiserson
3723:. pp. 145â171.
3416:
3362:
3279:
3199:
3132:
3065:
3017:
2938:
2927:{\displaystyle O(n)}
2909:
2888:
2856:
2749:
2664:
2493:
2317:
2297:
2245:
2139:
2067:
2028:{\displaystyle L(C)}
2010:
1999:As a consequence of
1935:
1715:
1608:
1165:
1091:
1071:
967:
895:
891:is the codeword for
868:
786:
687:
616:
594:
523:
475:Informal description
388:improve this article
271:variable-length code
5164:Compression formats
4803:Texture compression
4798:Standard test image
4614:Silence compression
3647:Scientific American
3356:ShannonâFano coding
3329:binary search trees
2714:A variation called
2679:{\displaystyle n=2}
2641:-ary Huffman coding
2611:Bernoulli processes
2607:run-length encoding
2332:{\displaystyle n-1}
1572:uniquely decodeable
342:ShannonâFano coding
5072:Information theory
4927:Display resolution
4753:Chroma subsampling
4142:Byte pair encoding
4087:ShannonâFanoâElias
3787:(1). IEEE: 27â38.
3452:
3398:
3303:
3252:
3185:
3118:
3051:
2962:
2924:
2894:
2874:
2823:
2761:
2738:commutative monoid
2676:
2512:
2487:canonical encoding
2329:
2303:
2251:
2231:
2169:
2125:
2057:
2025:
1986:
1960:
1918:
1879:
1810:
1755:
1666:
1439:Probability budget
1182:
1151:
1077:
1057:
945:
881:
854:
764:
673:
600:
580:
487:to probabilities).
472:
360:Problem definition
314:information theory
236:information theory
228:
5205:1952 in computing
5187:
5186:
5036:
5035:
4986:Deblocking filter
4884:
4883:
4732:
4731:
4541:
4540:
4386:
4385:
3827:10.1109/18.705558
3795:– via IEEE.
3611:978-3-319-05290-8
3471:Arithmetic coding
3325:Michelle L. Wachs
2897:{\displaystyle L}
2752:
2565:arithmetic coding
2306:{\displaystyle n}
2254:{\displaystyle n}
2223:
2222:
2135:with probability
1938:
1857:
1849:
1788:
1733:
1661:
1563:
1562:
1080:{\displaystyle C}
603:{\displaystyle n}
464:
463:
456:
438:
320:. The professor,
291:arithmetic coding
226:
225:
16:(Redirected from
5222:
5200:Data compression
5172:
5171:
5162:
5161:
4991:Lapped transform
4895:
4773:Image resolution
4758:Coding tree unit
4743:
4552:
4397:
4018:
4004:Data compression
3997:
3990:
3983:
3974:
3927:Ronald L. Rivest
3919:Thomas H. Cormen
3906:
3904:
3897:Knuth, Donald E.
3893:
3887:
3886:
3851:
3845:
3844:
3842:
3841:
3812:
3803:
3797:
3796:
3778:
3771:Karp, Richard M.
3767:
3761:
3760:
3732:
3719:(ONR). Salerno:
3708:
3702:
3701:
3679:
3673:
3672:
3670:
3658:
3652:
3651:
3637:
3631:
3622:
3616:
3615:
3595:
3589:
3588:
3586:
3585:
3575:
3567:Van Leeuwen, Jan
3563:
3557:
3556:
3547:(9): 1098â1101.
3536:
3524:
3461:
3459:
3458:
3453:
3412:, the result is
3407:
3405:
3404:
3399:
3312:
3310:
3309:
3304:
3261:
3259:
3258:
3253:
3251:
3247:
3223:
3219:
3194:
3192:
3191:
3186:
3184:
3180:
3156:
3152:
3127:
3125:
3124:
3119:
3117:
3113:
3089:
3085:
3060:
3058:
3057:
3052:
3050:
3046:
2971:
2969:
2968:
2963:
2933:
2931:
2930:
2925:
2903:
2901:
2900:
2895:
2883:
2881:
2880:
2875:
2832:
2830:
2829:
2824:
2822:
2818:
2817:
2813:
2812:
2799:
2776:
2775:
2760:
2685:
2683:
2682:
2677:
2525:
2521:
2519:
2518:
2513:
2511:
2510:
2338:
2336:
2335:
2330:
2312:
2310:
2309:
2304:
2260:
2258:
2257:
2252:
2181:
2178:
2176:
2175:
2170:
2134:
2132:
2131:
2126:
2121:
2120:
2108:
2107:
2095:
2094:
2082:
2081:
2034:
2032:
2031:
2026:
1995:
1993:
1992:
1987:
1973:
1972:
1959:
1958:
1957:
1927:
1925:
1924:
1919:
1914:
1913:
1912:
1899:
1898:
1889:
1888:
1878:
1871:
1870:
1850:
1848:
1847:
1835:
1830:
1829:
1820:
1819:
1809:
1802:
1801:
1781:
1780:
1765:
1764:
1754:
1747:
1746:
1707:
1696:
1675:
1673:
1672:
1667:
1662:
1660:
1659:
1647:
1642:
1641:
1626:
1625:
1533:
1484:
1444:
1405:
1395:
1362:
1342:
1337:
1332:
1327:
1322:
1316:
1276:
1242:
1218:
1191:
1189:
1188:
1183:
1181:
1160:
1158:
1157:
1152:
1150:
1146:
1145:
1120:
1116:
1115:
1086:
1084:
1083:
1078:
1066:
1064:
1063:
1058:
1056:
1055:
1051:
1050:
1031:
1030:
1019:
1014:
996:
992:
991:
954:
952:
951:
946:
907:
906:
890:
888:
887:
882:
880:
879:
863:
861:
860:
855:
850:
849:
831:
830:
818:
817:
802:
773:
771:
770:
765:
726:
722:
721:
699:
698:
682:
680:
679:
674:
669:
668:
650:
649:
637:
636:
609:
607:
606:
601:
589:
587:
586:
581:
576:
575:
557:
556:
544:
543:
459:
452:
448:
445:
439:
437:
403:"Huffman coding"
396:
372:
364:
307:David A. Huffman
279:entropy encoding
256:David A. Huffman
232:computer science
37:
21:
5232:
5231:
5225:
5224:
5223:
5221:
5220:
5219:
5190:
5189:
5188:
5183:
5150:
5134:
5118:
5099:Rateâdistortion
5032:
4961:
4880:
4807:
4728:
4633:
4629:Sub-band coding
4537:
4462:Predictive type
4457:
4382:
4349:LZSS + Huffman
4299:LZ77 + Huffman
4288:
4198:
4134:Dictionary type
4128:
4030:Adaptive coding
4007:
4001:
3954:
3915:
3910:
3909:
3895:
3894:
3890:
3875:10.1137/0121057
3853:
3852:
3848:
3839:
3837:
3810:
3805:
3804:
3800:
3769:
3768:
3764:
3749:
3730:10.1.1.589.4726
3710:
3709:
3705:
3681:
3680:
3676:
3660:
3659:
3655:
3639:
3638:
3634:
3623:
3619:
3612:
3597:
3596:
3592:
3583:
3581:
3573:
3565:
3564:
3560:
3534:
3526:
3525:
3521:
3516:
3468:
3414:
3413:
3360:
3359:
3343:
3337:
3277:
3276:
3266:problem, after
3231:
3227:
3209:
3205:
3197:
3196:
3164:
3160:
3142:
3138:
3130:
3129:
3097:
3093:
3075:
3071:
3063:
3062:
3030:
3026:
3015:
3014:
3010:
3002:Richard M. Karp
2979:
2936:
2935:
2907:
2906:
2886:
2885:
2854:
2853:
2839:
2804:
2800:
2767:
2766:
2762:
2747:
2746:
2735:totally ordered
2731:
2712:
2662:
2661:
2643:
2632:polynomial time
2627:
2552:
2546:
2538:frequency table
2533:
2531:Main properties
2523:
2502:
2491:
2490:
2482:
2315:
2314:
2313:leaf nodes and
2295:
2294:
2287:two child nodes
2243:
2242:
2137:
2136:
2112:
2099:
2086:
2073:
2065:
2064:
2053:
2046:
2041:
2039:Basic technique
2008:
2007:
1964:
1949:
1933:
1932:
1904:
1890:
1880:
1862:
1839:
1821:
1811:
1793:
1772:
1756:
1738:
1713:
1712:
1706:
1698:
1695:
1687:
1651:
1633:
1617:
1606:
1605:
1600:
1532:
1524:
1520:
1511:
1509:
1483:
1475:
1471:
1469:
1442:
1440:
1404:
1396:
1394:
1386:
1384:
1361:
1353:
1351:
1340:
1335:
1330:
1325:
1320:
1315:
1307:
1275:
1267:
1241:
1233:
1210:Shannon entropy
1198:
1171:
1163:
1162:
1135:
1131:
1127:
1105:
1101:
1097:
1089:
1088:
1069:
1068:
1042:
1038:
1022:
981:
977:
973:
965:
964:
962:
957:
956:
898:
893:
892:
871:
866:
865:
841:
822:
809:
792:
784:
783:
781:
776:
775:
713:
709:
690:
685:
684:
660:
641:
628:
614:
613:
611:
592:
591:
567:
548:
535:
521:
520:
518:
512:
477:
460:
449:
443:
440:
397:
395:
385:
373:
362:
350:
303:
258:while he was a
28:
23:
22:
15:
12:
11:
5:
5230:
5229:
5226:
5218:
5217:
5212:
5207:
5202:
5192:
5191:
5185:
5184:
5182:
5181:
5166:
5155:
5152:
5151:
5149:
5148:
5142:
5140:
5136:
5135:
5133:
5132:
5126:
5124:
5120:
5119:
5117:
5116:
5111:
5106:
5101:
5096:
5091:
5086:
5081:
5080:
5079:
5069:
5064:
5063:
5062:
5057:
5046:
5044:
5038:
5037:
5034:
5033:
5031:
5030:
5029:
5028:
5023:
5013:
5012:
5011:
5006:
5001:
4993:
4988:
4983:
4978:
4972:
4970:
4963:
4962:
4960:
4959:
4954:
4949:
4944:
4939:
4934:
4929:
4924:
4923:
4922:
4917:
4912:
4901:
4899:
4892:
4886:
4885:
4882:
4881:
4879:
4878:
4877:
4876:
4871:
4866:
4861:
4851:
4846:
4841:
4836:
4831:
4826:
4821:
4815:
4813:
4809:
4808:
4806:
4805:
4800:
4795:
4790:
4785:
4780:
4775:
4770:
4765:
4760:
4755:
4749:
4747:
4740:
4734:
4733:
4730:
4729:
4727:
4726:
4721:
4716:
4715:
4714:
4709:
4704:
4699:
4694:
4684:
4683:
4682:
4672:
4671:
4670:
4665:
4655:
4650:
4644:
4642:
4635:
4634:
4632:
4631:
4626:
4621:
4616:
4611:
4606:
4601:
4596:
4591:
4586:
4581:
4580:
4579:
4574:
4569:
4558:
4556:
4549:
4543:
4542:
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4538:
4536:
4535:
4533:Psychoacoustic
4530:
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4509:
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4503:
4498:
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4405:
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4401:Transform type
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3953:
3952:External links
3950:
3949:
3948:
3931:Clifford Stein
3914:
3911:
3908:
3907:
3888:
3846:
3798:
3773:(1961-01-31).
3762:
3747:
3703:
3692:(2): 228â230.
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3467:
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3367:
3339:Main article:
3336:
3333:
3321:Adriano Garsia
3302:
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3250:
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2730:
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2669:
2642:
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2623:
2585:block encoding
2545:
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2509:
2505:
2501:
2498:
2481:
2478:
2459:
2458:
2447:
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2428:
2422:
2419:
2395:) time, where
2373:
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2369:
2368:
2367:
2364:
2361:
2355:
2348:priority queue
2328:
2325:
2322:
2302:
2250:
2221:
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2209:
2205:
2204:
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2024:
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2018:
2015:
1985:
1982:
1979:
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1963:
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1598:
1588:Shannon (1948)
1586:As defined by
1561:
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491:
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461:
376:
374:
367:
361:
358:
349:
346:
338:Claude Shannon
322:Robert M. Fano
302:
299:
252:Huffman coding
224:
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220:
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26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5228:
5227:
5216:
5213:
5211:
5208:
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5137:
5131:
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5112:
5110:
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4989:
4987:
4984:
4982:
4979:
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4968:
4964:
4958:
4957:Video quality
4955:
4953:
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4908:
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4900:
4896:
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4887:
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4862:
4860:
4857:
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4852:
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4779:
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4678:
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4659:
4656:
4654:
4651:
4649:
4646:
4645:
4643:
4640:
4636:
4630:
4627:
4625:
4624:Speech coding
4622:
4620:
4619:Sound quality
4617:
4615:
4612:
4610:
4607:
4605:
4602:
4600:
4597:
4595:
4594:Dynamic range
4592:
4590:
4587:
4585:
4582:
4578:
4575:
4573:
4570:
4568:
4565:
4564:
4563:
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4507:
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4479:
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4452:
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4427:
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4412:
4411:
4410:
4407:
4406:
4404:
4402:
4398:
4395:
4393:
4389:
4377:
4374:
4373:
4371:
4366:
4364:
4361:
4360:
4359:LZ77 + Range
4358:
4354:
4351:
4350:
4348:
4344:
4341:
4340:
4338:
4334:
4331:
4330:
4328:
4324:
4321:
4320:
4318:
4314:
4311:
4309:
4306:
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4301:
4300:
4298:
4297:
4295:
4291:
4285:
4282:
4280:
4277:
4275:
4272:
4270:
4267:
4265:
4262:
4258:
4255:
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4250:
4249:
4248:
4245:
4243:
4240:
4238:
4235:
4231:
4228:
4227:
4226:
4223:
4221:
4218:
4216:
4213:
4211:
4208:
4207:
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4201:
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4185:
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4139:
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4135:
4131:
4123:
4120:
4118:
4115:
4113:
4110:
4108:
4105:
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4103:
4100:
4098:
4095:
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4090:
4088:
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4075:
4073:
4070:
4066:
4063:
4061:
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4056:
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4051:
4048:
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4043:
4041:
4038:
4036:
4033:
4031:
4028:
4027:
4025:
4023:
4019:
4016:
4014:
4010:
4005:
3998:
3993:
3991:
3986:
3984:
3979:
3978:
3975:
3969:
3966:
3964:
3961:
3959:
3956:
3955:
3951:
3946:
3945:0-262-03293-7
3942:
3938:
3937:
3932:
3928:
3924:
3920:
3917:
3916:
3912:
3902:
3898:
3892:
3889:
3884:
3880:
3876:
3872:
3868:
3864:
3860:
3859:Tucker, A. C.
3856:
3850:
3847:
3836:
3832:
3828:
3824:
3820:
3816:
3809:
3802:
3799:
3794:
3790:
3786:
3782:
3777:
3772:
3766:
3763:
3758:
3754:
3750:
3748:0-8186-8132-2
3744:
3740:
3736:
3731:
3726:
3722:
3718:
3714:
3707:
3704:
3699:
3695:
3691:
3687:
3686:
3678:
3675:
3669:
3664:
3657:
3654:
3649:
3648:
3643:
3636:
3633:
3629:
3628:
3621:
3618:
3613:
3607:
3603:
3602:
3594:
3591:
3579:
3572:
3568:
3562:
3559:
3554:
3550:
3546:
3542:
3541:
3533:
3529:
3523:
3520:
3513:
3511:
3509:
3505:
3501:
3497:
3493:
3489:
3485:
3480:
3477:
3472:
3465:
3463:
3446:
3443:
3440:
3437:
3434:
3431:
3428:
3425:
3422:
3411:
3392:
3389:
3386:
3383:
3380:
3377:
3374:
3371:
3368:
3357:
3353:
3349:
3342:
3334:
3332:
3330:
3326:
3322:
3318:
3314:
3297:
3294:
3291:
3288:
3282:
3273:
3269:
3265:
3248:
3244:
3241:
3238:
3235:
3232:
3228:
3224:
3220:
3216:
3213:
3210:
3206:
3202:
3181:
3177:
3174:
3171:
3168:
3165:
3161:
3157:
3153:
3149:
3146:
3143:
3139:
3135:
3114:
3110:
3107:
3104:
3101:
3098:
3094:
3090:
3086:
3082:
3079:
3076:
3072:
3068:
3047:
3043:
3040:
3037:
3034:
3031:
3027:
3023:
3020:
3007:
3005:
3003:
2999:
2994:
2990:
2988:
2984:
2976:
2974:
2972:
2956:
2953:
2950:
2947:
2941:
2918:
2912:
2891:
2868:
2865:
2859:
2851:
2847:
2843:
2836:
2834:
2819:
2814:
2809:
2805:
2801:
2777:
2772:
2768:
2763:
2757:
2743:
2739:
2736:
2728:
2726:
2724:
2719:
2718:
2709:
2707:
2705:
2701:
2697:
2693:
2689:
2673:
2670:
2667:
2659:
2655:
2651:
2649:
2640:
2637:
2635:
2633:
2624:
2622:
2620:
2616:
2615:Golomb coding
2612:
2608:
2603:
2599:
2597:
2592:
2590:
2586:
2582:
2577:
2574:
2568:
2566:
2562:
2558:
2551:
2543:
2541:
2539:
2530:
2528:
2507:
2503:
2499:
2496:
2488:
2480:Decompression
2479:
2477:
2473:
2471:
2467:
2462:
2456:
2452:
2448:
2445:
2444:
2443:
2437:
2432:
2429:
2426:
2425:
2423:
2420:
2417:
2416:
2415:
2413:
2409:
2405:
2400:
2398:
2394:
2390:
2386:
2382:
2378:
2370:
2365:
2362:
2359:
2358:
2356:
2353:
2352:
2351:
2349:
2344:
2340:
2326:
2323:
2320:
2300:
2292:
2288:
2284:
2280:
2276:
2272:
2268:
2267:internal node
2264:
2248:
2240:
2236:
2227:
2218:
2215:
2214:
2210:
2207:
2206:
2202:
2199:
2198:
2194:
2191:
2190:
2186:
2183:
2182:
2163:
2160:
2157:
2154:
2151:
2148:
2145:
2117:
2113:
2109:
2104:
2100:
2096:
2091:
2087:
2083:
2078:
2074:
2061:
2050:
2043:
2038:
2036:
2019:
2013:
2004:
2002:
1997:
1983:
1980:
1977:
1974:
1969:
1965:
1961:
1954:
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1943:
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1900:
1895:
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1858:
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1161:for any code
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1087:. Condition:
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444:December 2021
436:
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426:
422:
419:
415:
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405: â
404:
400:
399:Find sources:
393:
389:
383:
382:
377:This article
375:
371:
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365:
359:
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355:
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345:
343:
339:
334:
332:
327:
324:, assigned a
323:
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60:
56:
53:
50:
49:
45:
42:
39:
38:
32:
19:
5215:Binary trees
5130:Hutter Prize
5094:Quantization
4999:Compensation
4793:Quantization
4516:Compensation
4082:ShannonâFano
4049:
4022:Entropy type
3934:
3913:Bibliography
3900:
3891:
3866:
3862:
3849:
3838:. Retrieved
3818:
3814:
3801:
3784:
3780:
3765:
3712:
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3689:
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3677:
3656:
3645:
3635:
3626:
3620:
3600:
3593:
3582:. Retrieved
3577:
3561:
3544:
3538:
3522:
3508:quantization
3481:
3475:
3469:
3466:Applications
3409:
3351:
3347:
3344:
3263:
3011:
2992:
2991:
2986:
2982:
2980:
2841:
2840:
2741:
2732:
2715:
2713:
2703:
2699:
2695:
2691:
2687:
2657:
2653:
2650:-ary Huffman
2647:
2646:
2644:
2638:
2628:
2604:
2600:
2593:
2581:power of two
2578:
2569:
2553:
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2483:
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2465:
2463:
2460:
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2450:
2441:
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2388:
2384:
2383:leaves has 2
2380:
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2274:
2273:itself, the
2270:
2232:
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1301:
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1238:
1234:
1226:
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514:
513:
485:proportional
450:
441:
431:
424:
417:
410:
398:
386:Please help
381:verification
378:
351:
335:
304:
274:
268:
251:
240:Huffman code
239:
229:
18:Huffman code
5089:Prefix code
4942:Frame types
4763:Color space
4589:Convolution
4319:LZ77 + ANS
4230:Incremental
4203:Other types
4122:Levenshtein
3528:Huffman, D.
3272:Alan Tucker
2404:linear-time
2285:, links to
2235:binary tree
2044:Compression
1436:Optimality
1306:Codewords (
354:prefix code
348:Terminology
331:binary tree
262:student at
244:prefix code
5194:Categories
5146:Mark Adler
5104:Redundancy
5021:Daubechies
5004:Estimation
4937:Frame rate
4859:Daubechies
4819:Chain code
4778:Macroblock
4584:Companding
4521:Estimation
4441:Daubechies
4147:LempelâZiv
4107:Exp-Golomb
4035:Arithmetic
3869:(4): 514.
3840:2024-09-10
3668:1604.07476
3584:2014-02-20
3514:References
2998:Morse code
2625:Variations
2548:See also:
2544:Optimality
1559:) = 2.205
414:newspapers
326:term paper
5123:Community
4947:Interlace
4333:Zstandard
4112:Fibonacci
4102:Universal
4060:Canonical
3855:Hu, T. C.
3757:124587565
3725:CiteSeerX
3580:: 382â410
3295:
3264:HuâTucker
2954:
2500:⋅
2324:−
2263:leaf node
1975:
1947:→
1901:
1859:∑
1855:−
1832:
1790:∑
1735:∑
1644:
1431:) = 2.25
1266:Weights (
1122:≤
1036:
1002:∑
934:…
916:∈
836:…
753:…
735:∈
707:
655:…
562:…
519:Alphabet
305:In 1951,
5109:Symmetry
5077:Timeline
5060:FM-index
4905:Bit rate
4898:Concepts
4746:Concepts
4609:Sampling
4562:Bit rate
4555:Concepts
4257:Sequitur
4092:Tunstall
4065:Modified
4055:Adaptive
4013:Lossless
3650:: 54â58.
3569:(1976).
3530:(1952).
3498:such as
3268:T. C. Hu
2884:, where
2587:, e.g.,
2055:message.
1581:biunique
1577:complete
1567:biunique
1232:Symbol (
499:expected
309:and his
5067:Entropy
5016:Wavelet
4995:Motion
4854:Wavelet
4834:Fractal
4829:Deflate
4812:Methods
4599:Latency
4512:Motion
4436:Wavelet
4353:LHA/LZH
4303:Deflate
4252:Re-Pair
4247:Grammar
4077:Shannon
4050:Huffman
4006:methods
3883:2099603
3835:2265146
3488:Deflate
2226:entropy
1681:entropy
1503:
1463:= 1.00
1345:
1300:Output
1221:Input (
1196:Example
428:scholar
301:History
5178:codecs
5139:People
5042:Theory
5009:Vector
4526:Vector
4343:Brotli
4293:Hybrid
4192:Snappy
4045:Golomb
3943:
3929:, and
3881:
3833:
3755:
3745:
3727:
3608:
3496:codecs
2850:greedy
2596:dyadic
2573:patent
2412:queues
2291:parent
2283:weight
2279:parent
2275:weight
2271:symbol
2265:or an
2184:Symbol
1549:0.518
1546:0.423
1543:0.521
1540:0.411
1537:0.332
1033:length
778:Output
704:weight
612:Tuple
430:
423:
416:
409:
401:
283:linear
275:weight
222:10010
211:00111
200:11000
189:10011
178:00110
167:11001
4969:parts
4967:Codec
4932:Frame
4890:Video
4874:SPIHT
4783:Pixel
4738:Image
4692:ACELP
4663:ADPCM
4653:Îź-law
4648:A-law
4641:parts
4639:Codec
4547:Audio
4486:ACELP
4474:ADPCM
4451:SPIHT
4392:Lossy
4376:bzip2
4367:LZHAM
4323:LZFSE
4225:Delta
4117:Gamma
4097:Unary
4072:Range
3879:JSTOR
3831:S2CID
3811:(PDF)
3753:S2CID
3663:arXiv
3578:ICALP
3574:(PDF)
3535:(PDF)
3492:PKZIP
3313:-time
2589:ASCII
2239:array
2187:Code
1500:1.79
1497:2.64
1494:1.74
1491:2.74
1488:3.32
1421:0.58
1418:0.32
1415:0.60
1412:0.45
1409:0.30
1292:0.29
1289:0.16
1286:0.30
1283:0.15
1280:0.10
782:Code
515:Input
480:Given
435:JSTOR
421:books
260:Sc.D.
156:0110
145:1011
134:0010
123:0111
112:1000
101:1010
90:1101
51:space
46:Code
4981:DPCM
4788:PSNR
4719:MDCT
4712:WLPC
4697:CELP
4658:DPCM
4506:WLPC
4491:CELP
4469:DPCM
4419:MDCT
4363:LZMA
4264:LDCT
4242:DPCM
4187:LZWL
4177:LZSS
4172:LZRW
4162:LZJB
3941:ISBN
3743:ISBN
3721:IEEE
3606:ISBN
3502:and
3500:JPEG
3323:and
3270:and
2645:The
2391:log
2229:two.
2219:111
2211:110
2164:0.05
2152:0.35
1873:>
1804:>
1749:>
1679:The
1485:) â
1472:âlog
1460:1/4
1457:1/4
1454:1/4
1451:1/8
1448:1/8
1295:= 1
1261:Sum
963:Let
959:Goal
490:Find
407:news
318:exam
238:, a
234:and
79:000
68:010
57:111
43:Freq
40:Char
5026:DWT
4976:DCT
4920:VBR
4915:CBR
4910:ABR
4869:EZW
4864:DWT
4849:RLE
4839:KLT
4824:DCT
4707:LSP
4702:LAR
4687:LPC
4680:FFT
4577:VBR
4572:CBR
4567:ABR
4501:LSP
4496:LAR
4481:LPC
4446:DWT
4431:FFT
4426:DST
4414:DCT
4313:LZS
4308:LZX
4284:RLE
4279:PPM
4274:PAQ
4269:MTF
4237:DMC
4215:CTW
4210:BWT
4182:LZW
4167:LZO
4157:LZ4
4152:842
3871:doi
3823:doi
3789:doi
3735:doi
3694:doi
3549:doi
3504:MP3
3429:111
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2754:max
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2203:10
2158:0.2
2146:0.4
1966:log
1940:lim
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390:by
311:MIT
293:or
264:MIT
230:In
5196::
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54:7
20:)
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