4448:
4438:
2133:
2679:
2076:
in the second set receive "1". As long as any sets with more than one member remain, the same process is repeated on those sets, to determine successive digits of their codes. When a set has been reduced to one symbol this means the symbol's code is complete and will not form the prefix of any other symbol's code.
1929:
2075:
In Fano's method, the symbols are arranged in order from most probable to least probable, and then divided into two sets whose total probabilities are as close as possible to being equal. All symbols then have the first digits of their codes assigned; symbols in the first set receive "0" and symbols
2576:
is almost as computationally simple and produces prefix codes that always achieve the lowest possible expected code word length, under the constraints that each symbol is represented by a code formed of an integral number of bits. This is a constraint that is often unneeded, since the codes will be
142:
divides the source symbols into two sets ("0" and "1") with probabilities as close to 1/2 as possible. Then those sets are themselves divided in two, and so on, until each set contains only one symbol. The codeword for that symbol is the string of "0"s and "1"s that records which half of the divides
2757:
In this case D & E have the lowest frequencies and so are allocated 0 and 1 respectively and grouped together with a combined probability of 0.282. The lowest pair now are B and C so they're allocated 0 and 1 and grouped together with a combined probability of 0.333. This leaves BC and DE now
2079:
The algorithm produces fairly efficient variable-length encodings; when the two smaller sets produced by a partitioning are in fact of equal probability, the one bit of information used to distinguish them is used most efficiently. Unfortunately, Shannon–Fano coding does not always produce optimal
415:
Once the codeword lengths have been determined, we must choose the codewords themselves. One method is to pick codewords in order from most probable to least probable symbols, picking each codeword to be the lexicographically first word of the correct length that maintains the prefix-free property.
191:
There are several reasons for this mixup. For one thing, in the discussion of his coding scheme, Shannon mentions Fano’s scheme and calls it “substantially the same” (Shannon, 1948, p. 17 ). For another, both
Shannon’s and Fano’s coding schemes are similar in the sense that they both are efficient,
1127:
We can pick codewords in order, choosing the lexicographically first word of the correct length that maintains the prefix-free property. Clearly A gets the codeword 00. To maintain the prefix-free property, B's codeword may not start 00, so the lexicographically first available word of length 3 is
2215:
With this division, A and B will each have a code that starts with a 0 bit, and the C, D, and E codes will all start with a 1, as shown in Figure b. Subsequently, the left half of the tree gets a new division between A and B, which puts A on a leaf with code 00 and B on a leaf with code 01.
2622:
can produce greater overall compression than either
Huffman or Shannon–Fano, since it can encode in fractional numbers of bits which more closely approximate the actual information content of the symbol. However, arithmetic coding has not superseded Huffman the way that Huffman supersedes
2440:
1556:
2916:
2641:(1952) gave a different algorithm that always produces an optimal tree for any given symbol probabilities. While Fano's Shannon–Fano tree is created by dividing from the root to the leaves, the Huffman algorithm works in the opposite direction, merging from the leaves to the root.
183:
Around 1948, both Claude E. Shannon (1948) and Robert M. Fano (1949) independently proposed two different source coding algorithms for an efficient description of a discrete memoryless source. Unfortunately, in spite of being different, both schemes became known under the same name
3128:
The
Imploding algorithm is actually a combination of two distinct algorithms. The first algorithm compresses repeated byte sequences using a sliding dictionary. The second algorithm is used to compress the encoding of the sliding dictionary output, using multiple Shannon–Fano
2211:
All symbols are sorted by frequency, from left to right (shown in Figure a). Putting the dividing line between symbols B and C results in a total of 22 in the left group and a total of 17 in the right group. This minimizes the difference in totals between the two groups.
2758:
with the lowest probabilities so 0 and 1 are prepended to their codes and they are combined. This then leaves just A and BCDE, which have 0 and 1 prepended respectively and are then combined. This leaves us with a single node and our algorithm is complete.
1678:
159:
does. However, Shannon–Fano codes have an expected codeword length within 1 bit of optimal. Fano's method usually produces encoding with shorter expected lengths than
Shannon's method. However, Shannon's method is easier to analyse theoretically.
772:
This example shows the construction of a
Shannon–Fano code for a small alphabet. There 5 different source symbols. Suppose 39 total symbols have been observed with the following frequencies, from which we can estimate the symbol probabilities.
2119:
The left part of the list is assigned the binary digit 0, and the right part is assigned the digit 1. This means that the codes for the symbols in the first part will all start with 0, and the codes in the second part will all start with
2219:
After four division procedures, a tree of codes results. In the final tree, the three symbols with the highest frequencies have all been assigned 2-bit codes, and two symbols with lower counts have 3-bit codes as shown table below:
2336:
1452:
2818:
583:
1667:
2023:
2562:
2510:
684:
476:
945:
360:
126:
1446:
Note that although the codewords under the two methods are different, the word lengths are the same. We have lengths of 2 bits for A, and 3 bits for B, C, D and E, giving an average length of
1402:
1223:
1103:
298:
2123:
Recursively apply the steps 3 and 4 to each of the two halves, subdividing groups and adding bits to the codes until each symbol has become a corresponding code leaf on the tree.
2616:
1035:
386:
880:
2577:
packed end-to-end in long sequences. If we consider groups of codes at a time, symbol-by-symbol
Huffman coding is only optimal if the probabilities of the symbols are
1924:{\displaystyle \mathbb {E} L=\sum _{i=1}^{n}p_{i}l_{i}\leq \sum _{i=1}^{n}p_{i}(-\log _{2}p_{i}+1)=-\sum _{i=1}^{n}p_{i}\log _{2}p_{i}+\sum _{i=1}^{n}p_{i}=H(X)+1.}
762:
731:
2060:
704:
410:
64:
2939:
2080:
prefix codes; the set of probabilities {0.35, 0.17, 0.17, 0.16, 0.15} is an example of one that will be assigned non-optimal codes by
Shannon–Fano coding.
484:
2435:{\displaystyle {\frac {2\,{\text{bits}}\cdot (15+7+6)+3\,{\text{bits}}\cdot (6+5)}{39\,{\text{symbols}}}}\approx 2.28\,{\text{bits per symbol.}}}
1551:{\displaystyle {\frac {2\,{\text{bits}}\cdot (15)+3\,{\text{bits}}\cdot (7+6+6+5)}{39\,{\text{symbols}}}}\approx 2.62\,{\text{bits per symbol,}}}
1575:
128:. One common way of choosing the codewords uses the binary expansion of the cumulative probabilities. This method was proposed in Shannon's "
2911:{\displaystyle {\frac {1\,{\text{bit}}\cdot 15+3\,{\text{bits}}\cdot (7+6+6+5)}{39\,{\text{symbols}}}}\approx 2.23\,{\text{bits per symbol.}}}
3272:
2113:
Sort the lists of symbols according to frequency, with the most frequently occurring symbols at the left and the least common at the right.
3940:
3751:
1937:
2116:
Divide the list into two parts, with the total frequency counts of the left part being as close to the total of the right as possible.
2102:
A Shannon–Fano tree is built according to a specification designed to define an effective code table. The actual algorithm is simple:
3640:
2572:
Neither
Shannon–Fano algorithm is guaranteed to generate an optimal code. For this reason, Shannon–Fano codes are almost never used;
4146:
3969:
3763:
2661:
Create a new internal node with these two nodes as children and with probability equal to the sum of the two nodes' probabilities.
3454:
4151:
3728:
129:
2623:
Shannon–Fano, both because arithmetic coding is more computationally expensive and because it is covered by multiple patents.
2515:
2445:
We see that Fano's method, with an average length of 2.28, has outperformed
Shannon's method, with an average length of 2.62.
3152:
2456:
2921:
We see that the
Huffman code has outperformed both types of Shannon–Fano code, which had expected lengths of 2.62 and 2.28.
3881:
591:
422:
238:
Shannon's method starts by deciding on the lengths of all the codewords, then picks a prefix code with those word lengths.
3172:
888:
303:
69:
4258:
3996:
3935:
3746:
3696:
3519:
2030:
3379:
3364:
3265:
164:
4371:
1358:
1179:
1059:
4381:
4219:
4070:
3989:
3783:
2991:
Stanislav Krajči, Chin-Fu Liu, Ladislav Mikeš and Stefan M. Moser (2015), "Performance analysis of Fano coding",
2962:
419:
A second method makes use of cumulative probabilities. First, the probabilities are written in decreasing order
4354:
3974:
3768:
3556:
3005:
244:
4451:
3487:
3230:
2761:
The code lengths for the different characters this time are 1 bit for A and 3 bits for all other characters.
4116:
179:
Regarding the confusion in the two different codes being referred to by the same name, Krajči et al. write:
4441:
4344:
3886:
3444:
3258:
2453:
It is shown by Krajči et al that the expected length of Fano's method has expected length bounded above by
2026:
845:
3434:
3429:
2812:
This results in the lengths of 1 bit for A and per 3 bits for B, C, D and E, giving an average length of
2330:
This results in lengths of 2 bits for A, B and C and per 3 bits for D and E, giving an average length of
389:
2584:
2062:. Hence we see that the Shannon–Fano code is always within one bit of the optimal expected word length.
4376:
4303:
4141:
4121:
4065:
3723:
3514:
3317:
152:
4477:
4386:
4327:
4253:
4101:
3691:
3686:
3541:
3384:
997:
365:
4391:
3964:
3758:
3459:
2578:
4487:
4482:
4332:
3703:
3590:
3546:
3342:
3332:
850:
3957:
3708:
3492:
3337:
3180:
4229:
155:
in the sense that they do not always achieve the lowest possible expected codeword length, as
4361:
204:
by Cover and Thomas, Goldie and Pinch, Jones and Jones, and Han and Kobayashi. It is called
4045:
3507:
3469:
3290:
740:
709:
3225:
3112:
2036:
8:
4276:
4167:
4126:
4111:
4080:
4075:
3984:
3891:
3824:
3793:
3778:
3561:
4349:
4319:
4298:
4204:
4136:
4030:
3718:
3534:
3524:
3419:
3399:
3394:
3243:
2954:
689:
395:
133:
49:
3930:
4293:
4281:
4263:
4131:
4015:
3952:
3798:
3713:
3669:
3630:
3312:
3148:
2958:
2619:
2088:
168:
2110:
or frequency counts so that each symbol’s relative frequency of occurrence is known.
4268:
4224:
4197:
4192:
4050:
4035:
3945:
3854:
3849:
3678:
3411:
3389:
3281:
3239:
3189:
3168:
2638:
144:
20:
4187:
4001:
3925:
3906:
3876:
3844:
3810:
3369:
3307:
3142:
578:{\displaystyle c_{1}=0,\qquad c_{i}=\sum _{j=1}^{i-1}p_{j}{\text{ for }}i\geq 2,}
3979:
3773:
3502:
3497:
3354:
3327:
3299:
3193:
2646:
2632:
2573:
228:
156:
28:
4471:
4286:
4234:
3901:
3896:
3871:
3803:
3424:
3322:
2107:
734:
4407:
3374:
3349:
3250:
3226:"A Mathematical Theory of Communication [reprint with corrections]"
3212:
39:
based on a set of symbols and their probabilities (estimated or measured).
1662:{\displaystyle l_{i}=\lceil -\log _{2}p_{i}\rceil \leq -\log _{2}p_{i}+1.}
4366:
4244:
4040:
3916:
3866:
885:
For the Shannon–Fano code, we need to calculate the desired word lengths
36:
32:
2686:
We use the same frequencies as for the Shannon–Fano example above, viz:
2658:
Prepend 0 and 1 respectively to any code already assigned to these nodes
211:
Fano's (1949) method, using binary division of probabilities, is called
4423:
4214:
4209:
4096:
4055:
3861:
2655:
Remove the two nodes of lowest probability or frequency from the queue
3219:. Cambridge (Mass.), USA: Research Laboratory of Electronics at MIT.
3140:
4337:
4182:
3839:
2018:{\displaystyle H(X)=-\textstyle \sum _{i=1}^{n}p_{i}\log _{2}p_{i}}
200:
Shannon's (1948) method, using predefined word lengths, is called
4106:
3580:
3529:
2678:
2132:
3620:
3024:(2nd ed.), Wiley–Interscience. "Historical Notes" to Chapter 5.
2993:
2015 IEEE International Symposium on Information Theory (ISIT)
4455:
4060:
3653:
3600:
2669:
The remaining node is the root node and the tree is complete.
2106:
For a given list of symbols, develop a corresponding list of
1267:
Alternatively, we can use the cumulative probability method.
3610:
3464:
3449:
3439:
3173:"A Method for the Construction of Minimum-Redundancy Codes"
2947:
Journal of Network Communications and Emerging Technologies
2033:
says that any code must have an average length of at least
143:
it fell on. This method was proposed in a later (in print)
222:
3585:
3551:
3141:
Ze-Nian Li; Mark S. Drew; Jiangchuan Liu (9 April 2014).
3010:. AT & T Bell Laboratories. 1948-07-01. p. 403.
2557:{\displaystyle p_{\text{min}}=\textstyle \min _{i}p_{i}}
392:, meaning the smallest integer greater than or equal to
3007:
The Bell System Technical Journal 1948-07: Vol 27 Iss 3
2567:
2505:{\displaystyle \mathbb {E} L\leq H(X)+1-p_{\text{min}}}
196:
prefix-free coding schemes with a similar performance.
2588:
2532:
1959:
1128:
010. Continuing like this, we get the following code:
35:, is one of two related techniques for constructing a
2821:
2587:
2518:
2459:
2339:
2083:
Fano's version of Shannon–Fano coding is used in the
2039:
1940:
1681:
1578:
1455:
1361:
1182:
1062:
1000:
891:
853:
743:
712:
692:
679:{\displaystyle c_{1}=0,c_{2}=p_{1},c_{3}=p_{1}+p_{2}}
594:
487:
471:{\displaystyle p_{1}\geq p_{2}\geq \cdots \geq p_{n}}
425:
398:
368:
306:
247:
72:
52:
2649:, using its frequency of occurrence as the priority.
478:. Then, the cumulative probabilities are defined as
2645:Create a leaf node for each symbol and add it to a
940:{\displaystyle l_{i}=\lceil -\log _{2}p_{i}\rceil }
355:{\displaystyle l_{i}=\lceil -\log _{2}p_{i}\rceil }
121:{\displaystyle l_{i}=\lceil -\log _{2}p_{i}\rceil }
3063:, American Mathematical Society. Subsection 3.7.1.
3033:Charles M. Goldie and Richard G. E. Pinch (1991),
2910:
2610:
2556:
2504:
2434:
2054:
2017:
1923:
1661:
1550:
1396:
1217:
1097:
1029:
939:
874:
756:
725:
698:
678:
577:
470:
404:
380:
354:
292:
120:
58:
4469:
2940:"Entropy Coding and Different Coding Techniques"
2652:While there is more than one node in the queue:
2534:
2065:
163:Shannon–Fano coding should not be confused with
2564:is the probability of the least common symbol.
1569:For Shannon's method, the word lengths satisfy
167:(also known as Elias coding), the precursor to
132:" (1948), his article introducing the field of
2673:
3266:
1397:{\displaystyle \lceil -\log _{2}p_{i}\rceil }
1218:{\displaystyle \lceil -\log _{2}p_{i}\rceil }
1098:{\displaystyle \lceil -\log _{2}p_{i}\rceil }
3280:
1621:
1592:
1391:
1362:
1212:
1183:
1092:
1063:
934:
905:
375:
369:
349:
320:
115:
86:
46:chooses a prefix code where a source symbol
2938:Kaur, Sandeep; Singh, Sukhjeet (May 2016).
3273:
3259:
3134:
3046:Gareth A. Jones and J. Mary Jones (2012),
3037:, Cambridge University Press. Section 1.6.
3020:Thomas M. Cover and Joy A. Thomas (2006),
3147:. Springer Science & Business Media.
3100:Data Communications and Computer Networks
2937:
2902:
2887:
2846:
2828:
2461:
2448:
2426:
2411:
2382:
2346:
2087:compression method, which is part of the
2070:
1683:
1672:Hence the expected word length satisfies
1542:
1527:
1486:
1462:
3087:Data Compression: The Complete Reference
2677:
2131:
2097:
1561:which is within one bit of the entropy.
293:{\displaystyle p_{1},p_{2},\dots ,p_{n}}
3223:
3167:
3105:
3059:Te Sun Han and Kingo Kobayashi (2007),
2140:We continue with the previous example.
1564:
223:Shannon's code: predefined word lengths
4470:
2987:
2985:
2983:
233:
130:A Mathematical Theory of Communication
3254:
3061:Mathematics of Information and Coding
3210:
3102:, Phi Publishing. Subsection 1.11.5.
3074:A First Course in Information Theory
2581:and are some power of a half, i.e.,
2568:Comparison with other coding methods
2980:
2931:
686:and so on. The codeword for symbol
215:by Salomon and Gupta. It is called
13:
3244:10.1002/j.1538-7305.1948.tb01338.x
2611:{\displaystyle \textstyle 1/2^{k}}
241:Given a source with probabilities
14:
4499:
3213:"The transmission of information"
3117:- .ZIP File Format Specification"
2626:
300:the desired codeword lengths are
4447:
4446:
4437:
4436:
3161:
3092:
2031:Shannon's source coding theorem
1030:{\displaystyle -\log _{2}p_{i}}
507:
381:{\displaystyle \lceil x\rceil }
3079:
3066:
3053:
3040:
3027:
3022:Elements of Information Theory
3014:
2998:
2879:
2855:
2664:Add the new node to the queue.
2480:
2474:
2403:
2391:
2373:
2355:
2049:
2043:
1950:
1944:
1912:
1906:
1803:
1768:
1519:
1495:
1477:
1471:
863:
857:
1:
3231:Bell System Technical Journal
3204:
3076:, Springer. Subsection 3.2.2.
3048:Information and Coding Theory
2066:Fano's code: binary splitting
66:is given the codeword length
2127:
7:
3224:Shannon, C.E. (July 1948).
2674:Example with Huffman coding
16:Data compression algorithms
10:
4504:
4328:Compressed data structures
3650:RLE + BWT + MTF + Huffman
3318:Asymmetric numeral systems
3194:10.1109/JRPROC.1952.273898
3144:Fundamentals of Multimedia
2630:
875:{\displaystyle H(X)=2.186}
767:
706:is chosen to be the first
226:
4432:
4416:
4400:
4318:
4243:
4175:
4166:
4089:
4023:
4014:
3915:
3832:
3823:
3739:
3687:Discrete cosine transform
3677:
3668:
3617:LZ77 + Huffman + context
3570:
3480:
3410:
3298:
3289:
3098:Prakash C. Gupta (2006),
2290:
2287:
2284:
2281:
2273:
2270:
1315:Cumulative probabilities
174:
165:Shannon–Fano–Elias coding
4392:Smallest grammar problem
3119:. PKWARE Inc. 2007-09-28
3089:, Springer. Section 2.6.
3072:Raymond W Yeung (2002),
3050:(Springer). Section 3.4.
2924:
4333:Compressed suffix array
3882:Nyquist–Shannon theorem
3217:Technical Report No. 65
2618:. In most situations,
151:Shannon–Fano codes are
3181:Proceedings of the IRE
3085:David Salomon (2013),
2912:
2683:
2612:
2558:
2506:
2436:
2137:
2136:Shannon–Fano Algorithm
2071:Outline of Fano's code
2056:
2019:
1980:
1925:
1889:
1832:
1757:
1713:
1663:
1552:
1398:
1219:
1099:
1031:
941:
876:
758:
727:
700:
680:
579:
547:
472:
406:
382:
356:
294:
198:
122:
60:
4362:Kolmogorov complexity
4230:Video characteristics
3607:LZ77 + Huffman + ANS
2913:
2681:
2613:
2559:
2507:
2437:
2135:
2098:The Shannon–Fano tree
2057:
2020:
1960:
1926:
1869:
1812:
1737:
1693:
1664:
1553:
1399:
1220:
1100:
1032:
942:
877:
759:
757:{\displaystyle c_{i}}
733:binary digits in the
728:
726:{\displaystyle l_{i}}
701:
681:
580:
521:
473:
407:
383:
357:
295:
181:
123:
61:
4452:Compression software
4046:Compression artifact
4002:Psychoacoustic model
3035:Communication Theory
2819:
2585:
2516:
2457:
2449:Expected word length
2337:
2055:{\displaystyle H(X)}
2037:
1938:
1679:
1576:
1565:Expected word length
1453:
1359:
1180:
1060:
998:
889:
851:
741:
710:
690:
592:
485:
423:
396:
366:
304:
245:
70:
50:
4442:Compression formats
4081:Texture compression
4076:Standard test image
3892:Silence compression
3211:Fano, R.M. (1949).
2637:A few years later,
234:Shannon's algorithm
213:Shannon–Fano coding
202:Shannon–Fano coding
186:Shannon–Fano coding
25:Shannon–Fano coding
4350:Information theory
4205:Display resolution
4031:Chroma subsampling
3420:Byte pair encoding
3365:Shannon–Fano–Elias
2908:
2684:
2608:
2607:
2554:
2553:
2542:
2502:
2432:
2138:
2052:
2015:
2014:
1921:
1659:
1548:
1394:
1215:
1095:
1027:
937:
872:
754:
723:
696:
676:
575:
468:
402:
378:
352:
290:
134:information theory
118:
56:
4465:
4464:
4314:
4313:
4264:Deblocking filter
4162:
4161:
4010:
4009:
3819:
3818:
3664:
3663:
3154:978-3-319-05290-8
2906:
2894:
2891:
2850:
2832:
2808:
2807:
2753:
2752:
2682:Huffman Algorithm
2620:arithmetic coding
2533:
2526:
2499:
2430:
2418:
2415:
2386:
2350:
2326:
2325:
2207:
2206:
1546:
1534:
1531:
1490:
1466:
1442:
1441:
1263:
1262:
1123:
1122:
840:
839:
699:{\displaystyle i}
561:
405:{\displaystyle x}
219:by Krajči et al.
169:arithmetic coding
59:{\displaystyle i}
4495:
4478:Data compression
4450:
4449:
4440:
4439:
4269:Lapped transform
4173:
4172:
4051:Image resolution
4036:Coding tree unit
4021:
4020:
3830:
3829:
3675:
3674:
3296:
3295:
3282:Data compression
3275:
3268:
3261:
3252:
3251:
3247:
3220:
3198:
3197:
3188:(9): 1098–1101.
3177:
3165:
3159:
3158:
3138:
3132:
3131:
3125:
3124:
3116:
3109:
3103:
3096:
3090:
3083:
3077:
3070:
3064:
3057:
3051:
3044:
3038:
3031:
3025:
3018:
3012:
3011:
3002:
2996:
2989:
2978:
2977:
2975:
2973:
2967:
2961:. Archived from
2944:
2935:
2917:
2915:
2914:
2909:
2907:
2905:bits per symbol.
2904:
2895:
2893:
2892:
2889:
2882:
2851:
2848:
2833:
2830:
2823:
2766:
2765:
2691:
2690:
2639:David A. Huffman
2617:
2615:
2614:
2609:
2606:
2605:
2596:
2563:
2561:
2560:
2555:
2552:
2551:
2541:
2528:
2527:
2524:
2511:
2509:
2508:
2503:
2501:
2500:
2497:
2464:
2441:
2439:
2438:
2433:
2431:
2429:bits per symbol.
2428:
2419:
2417:
2416:
2413:
2406:
2387:
2384:
2351:
2348:
2341:
2279:Second division
2225:
2224:
2145:
2144:
2091:
2086:
2061:
2059:
2058:
2053:
2024:
2022:
2021:
2016:
2013:
2012:
2000:
1999:
1990:
1989:
1979:
1974:
1930:
1928:
1927:
1922:
1899:
1898:
1888:
1883:
1865:
1864:
1852:
1851:
1842:
1841:
1831:
1826:
1796:
1795:
1783:
1782:
1767:
1766:
1756:
1751:
1733:
1732:
1723:
1722:
1712:
1707:
1686:
1668:
1666:
1665:
1660:
1652:
1651:
1639:
1638:
1620:
1619:
1607:
1606:
1588:
1587:
1557:
1555:
1554:
1549:
1547:
1545:bits per symbol,
1544:
1535:
1533:
1532:
1529:
1522:
1491:
1488:
1467:
1464:
1457:
1403:
1401:
1400:
1395:
1390:
1389:
1377:
1376:
1272:
1271:
1224:
1222:
1221:
1216:
1211:
1210:
1198:
1197:
1133:
1132:
1104:
1102:
1101:
1096:
1091:
1090:
1078:
1077:
1036:
1034:
1033:
1028:
1026:
1025:
1013:
1012:
952:
951:
946:
944:
943:
938:
933:
932:
920:
919:
901:
900:
881:
879:
878:
873:
844:This source has
778:
777:
763:
761:
760:
755:
753:
752:
735:binary expansion
732:
730:
729:
724:
722:
721:
705:
703:
702:
697:
685:
683:
682:
677:
675:
674:
662:
661:
649:
648:
636:
635:
623:
622:
604:
603:
584:
582:
581:
576:
562:
559:
557:
556:
546:
535:
517:
516:
497:
496:
477:
475:
474:
469:
467:
466:
448:
447:
435:
434:
411:
409:
408:
403:
390:ceiling function
387:
385:
384:
379:
361:
359:
358:
353:
348:
347:
335:
334:
316:
315:
299:
297:
296:
291:
289:
288:
270:
269:
257:
256:
145:technical report
127:
125:
124:
119:
114:
113:
101:
100:
82:
81:
65:
63:
62:
57:
44:Shannon's method
21:data compression
19:In the field of
4503:
4502:
4498:
4497:
4496:
4494:
4493:
4492:
4468:
4467:
4466:
4461:
4428:
4412:
4396:
4377:Rate–distortion
4310:
4239:
4158:
4085:
4006:
3911:
3907:Sub-band coding
3815:
3740:Predictive type
3735:
3660:
3627:LZSS + Huffman
3577:LZ77 + Huffman
3566:
3476:
3412:Dictionary type
3406:
3308:Adaptive coding
3285:
3279:
3207:
3202:
3201:
3175:
3166:
3162:
3155:
3139:
3135:
3122:
3120:
3114:
3111:
3110:
3106:
3097:
3093:
3084:
3080:
3071:
3067:
3058:
3054:
3045:
3041:
3032:
3028:
3019:
3015:
3004:
3003:
2999:
2990:
2981:
2971:
2969:
2965:
2942:
2936:
2932:
2927:
2903:
2888:
2883:
2847:
2829:
2824:
2822:
2820:
2817:
2816:
2676:
2635:
2629:
2601:
2597:
2592:
2586:
2583:
2582:
2570:
2547:
2543:
2537:
2523:
2519:
2517:
2514:
2513:
2496:
2492:
2460:
2458:
2455:
2454:
2451:
2427:
2412:
2407:
2383:
2347:
2342:
2340:
2338:
2335:
2334:
2296:Third division
2268:First division
2130:
2100:
2089:
2084:
2073:
2068:
2038:
2035:
2034:
2008:
2004:
1995:
1991:
1985:
1981:
1975:
1964:
1939:
1936:
1935:
1894:
1890:
1884:
1873:
1860:
1856:
1847:
1843:
1837:
1833:
1827:
1816:
1791:
1787:
1778:
1774:
1762:
1758:
1752:
1741:
1728:
1724:
1718:
1714:
1708:
1697:
1682:
1680:
1677:
1676:
1647:
1643:
1634:
1630:
1615:
1611:
1602:
1598:
1583:
1579:
1577:
1574:
1573:
1567:
1543:
1528:
1523:
1487:
1463:
1458:
1456:
1454:
1451:
1450:
1385:
1381:
1372:
1368:
1360:
1357:
1356:
1206:
1202:
1193:
1189:
1181:
1178:
1177:
1086:
1082:
1073:
1069:
1061:
1058:
1057:
1021:
1017:
1008:
1004:
999:
996:
995:
928:
924:
915:
911:
896:
892:
890:
887:
886:
852:
849:
848:
770:
748:
744:
742:
739:
738:
717:
713:
711:
708:
707:
691:
688:
687:
670:
666:
657:
653:
644:
640:
631:
627:
618:
614:
599:
595:
593:
590:
589:
560: for
558:
552:
548:
536:
525:
512:
508:
492:
488:
486:
483:
482:
462:
458:
443:
439:
430:
426:
424:
421:
420:
397:
394:
393:
367:
364:
363:
343:
339:
330:
326:
311:
307:
305:
302:
301:
284:
280:
265:
261:
252:
248:
246:
243:
242:
236:
231:
225:
177:
147:by Fano (1949).
109:
105:
96:
92:
77:
73:
71:
68:
67:
51:
48:
47:
17:
12:
11:
5:
4501:
4491:
4490:
4488:Entropy coding
4485:
4483:Claude Shannon
4480:
4463:
4462:
4460:
4459:
4444:
4433:
4430:
4429:
4427:
4426:
4420:
4418:
4414:
4413:
4411:
4410:
4404:
4402:
4398:
4397:
4395:
4394:
4389:
4384:
4379:
4374:
4369:
4364:
4359:
4358:
4357:
4347:
4342:
4341:
4340:
4335:
4324:
4322:
4316:
4315:
4312:
4311:
4309:
4308:
4307:
4306:
4301:
4291:
4290:
4289:
4284:
4279:
4271:
4266:
4261:
4256:
4250:
4248:
4241:
4240:
4238:
4237:
4232:
4227:
4222:
4217:
4212:
4207:
4202:
4201:
4200:
4195:
4190:
4179:
4177:
4170:
4164:
4163:
4160:
4159:
4157:
4156:
4155:
4154:
4149:
4144:
4139:
4129:
4124:
4119:
4114:
4109:
4104:
4099:
4093:
4091:
4087:
4086:
4084:
4083:
4078:
4073:
4068:
4063:
4058:
4053:
4048:
4043:
4038:
4033:
4027:
4025:
4018:
4012:
4011:
4008:
4007:
4005:
4004:
3999:
3994:
3993:
3992:
3987:
3982:
3977:
3972:
3962:
3961:
3960:
3950:
3949:
3948:
3943:
3933:
3928:
3922:
3920:
3913:
3912:
3910:
3909:
3904:
3899:
3894:
3889:
3884:
3879:
3874:
3869:
3864:
3859:
3858:
3857:
3852:
3847:
3836:
3834:
3827:
3821:
3820:
3817:
3816:
3814:
3813:
3811:Psychoacoustic
3808:
3807:
3806:
3801:
3796:
3788:
3787:
3786:
3781:
3776:
3771:
3766:
3756:
3755:
3754:
3743:
3741:
3737:
3736:
3734:
3733:
3732:
3731:
3726:
3721:
3711:
3706:
3701:
3700:
3699:
3694:
3683:
3681:
3679:Transform type
3672:
3666:
3665:
3662:
3661:
3659:
3658:
3657:
3656:
3648:
3647:
3646:
3643:
3635:
3634:
3633:
3625:
3624:
3623:
3615:
3614:
3613:
3605:
3604:
3603:
3595:
3594:
3593:
3588:
3583:
3574:
3572:
3568:
3567:
3565:
3564:
3559:
3554:
3549:
3544:
3539:
3538:
3537:
3532:
3522:
3517:
3512:
3511:
3510:
3500:
3495:
3490:
3484:
3482:
3478:
3477:
3475:
3474:
3473:
3472:
3467:
3462:
3457:
3452:
3447:
3442:
3437:
3432:
3422:
3416:
3414:
3408:
3407:
3405:
3404:
3403:
3402:
3397:
3392:
3387:
3377:
3372:
3367:
3362:
3357:
3352:
3347:
3346:
3345:
3340:
3335:
3325:
3320:
3315:
3310:
3304:
3302:
3293:
3287:
3286:
3278:
3277:
3270:
3263:
3255:
3249:
3248:
3221:
3206:
3203:
3200:
3199:
3160:
3153:
3133:
3104:
3091:
3078:
3065:
3052:
3039:
3026:
3013:
2997:
2979:
2929:
2928:
2926:
2923:
2919:
2918:
2901:
2898:
2886:
2881:
2878:
2875:
2872:
2869:
2866:
2863:
2860:
2857:
2854:
2845:
2842:
2839:
2836:
2827:
2810:
2809:
2806:
2805:
2802:
2799:
2796:
2793:
2790:
2786:
2785:
2782:
2779:
2776:
2773:
2770:
2755:
2754:
2751:
2750:
2747:
2744:
2741:
2738:
2735:
2734:Probabilities
2731:
2730:
2727:
2724:
2721:
2718:
2715:
2711:
2710:
2707:
2704:
2701:
2698:
2695:
2675:
2672:
2671:
2670:
2667:
2666:
2665:
2662:
2659:
2656:
2650:
2647:priority queue
2633:Huffman coding
2631:Main article:
2628:
2627:Huffman coding
2625:
2604:
2600:
2595:
2591:
2574:Huffman coding
2569:
2566:
2550:
2546:
2540:
2536:
2531:
2522:
2495:
2491:
2488:
2485:
2482:
2479:
2476:
2473:
2470:
2467:
2463:
2450:
2447:
2443:
2442:
2425:
2422:
2410:
2405:
2402:
2399:
2396:
2393:
2390:
2381:
2378:
2375:
2372:
2369:
2366:
2363:
2360:
2357:
2354:
2345:
2328:
2327:
2324:
2323:
2320:
2317:
2314:
2311:
2308:
2304:
2303:
2300:
2297:
2293:
2292:
2289:
2286:
2283:
2280:
2276:
2275:
2272:
2269:
2265:
2264:
2261:
2258:
2255:
2252:
2249:
2248:Probabilities
2245:
2244:
2241:
2238:
2235:
2232:
2229:
2209:
2208:
2205:
2204:
2201:
2198:
2195:
2192:
2189:
2188:Probabilities
2185:
2184:
2181:
2178:
2175:
2172:
2169:
2165:
2164:
2161:
2158:
2155:
2152:
2149:
2129:
2126:
2125:
2124:
2121:
2117:
2114:
2111:
2099:
2096:
2072:
2069:
2067:
2064:
2051:
2048:
2045:
2042:
2011:
2007:
2003:
1998:
1994:
1988:
1984:
1978:
1973:
1970:
1967:
1963:
1958:
1955:
1952:
1949:
1946:
1943:
1932:
1931:
1920:
1917:
1914:
1911:
1908:
1905:
1902:
1897:
1893:
1887:
1882:
1879:
1876:
1872:
1868:
1863:
1859:
1855:
1850:
1846:
1840:
1836:
1830:
1825:
1822:
1819:
1815:
1811:
1808:
1805:
1802:
1799:
1794:
1790:
1786:
1781:
1777:
1773:
1770:
1765:
1761:
1755:
1750:
1747:
1744:
1740:
1736:
1731:
1727:
1721:
1717:
1711:
1706:
1703:
1700:
1696:
1692:
1689:
1685:
1670:
1669:
1658:
1655:
1650:
1646:
1642:
1637:
1633:
1629:
1626:
1623:
1618:
1614:
1610:
1605:
1601:
1597:
1594:
1591:
1586:
1582:
1566:
1563:
1559:
1558:
1541:
1538:
1526:
1521:
1518:
1515:
1512:
1509:
1506:
1503:
1500:
1497:
1494:
1485:
1482:
1479:
1476:
1473:
1470:
1461:
1444:
1443:
1440:
1439:
1436:
1433:
1430:
1427:
1424:
1420:
1419:
1416:
1413:
1410:
1407:
1404:
1393:
1388:
1384:
1380:
1375:
1371:
1367:
1364:
1352:
1351:
1348:
1345:
1342:
1339:
1336:
1332:
1331:
1328:
1325:
1322:
1319:
1316:
1312:
1311:
1308:
1305:
1302:
1299:
1296:
1295:Probabilities
1292:
1291:
1288:
1285:
1282:
1279:
1276:
1265:
1264:
1261:
1260:
1257:
1254:
1251:
1248:
1245:
1241:
1240:
1237:
1234:
1231:
1228:
1225:
1214:
1209:
1205:
1201:
1196:
1192:
1188:
1185:
1173:
1172:
1169:
1166:
1163:
1160:
1157:
1156:Probabilities
1153:
1152:
1149:
1146:
1143:
1140:
1137:
1125:
1124:
1121:
1120:
1117:
1114:
1111:
1108:
1105:
1094:
1089:
1085:
1081:
1076:
1072:
1068:
1065:
1053:
1052:
1049:
1046:
1043:
1040:
1037:
1024:
1020:
1016:
1011:
1007:
1003:
992:
991:
988:
985:
982:
979:
976:
975:Probabilities
972:
971:
968:
965:
962:
959:
956:
936:
931:
927:
923:
918:
914:
910:
907:
904:
899:
895:
871:
868:
865:
862:
859:
856:
842:
841:
838:
837:
834:
831:
828:
825:
822:
821:Probabilities
818:
817:
814:
811:
808:
805:
802:
798:
797:
794:
791:
788:
785:
782:
769:
766:
751:
747:
720:
716:
695:
673:
669:
665:
660:
656:
652:
647:
643:
639:
634:
630:
626:
621:
617:
613:
610:
607:
602:
598:
586:
585:
574:
571:
568:
565:
555:
551:
545:
542:
539:
534:
531:
528:
524:
520:
515:
511:
506:
503:
500:
495:
491:
465:
461:
457:
454:
451:
446:
442:
438:
433:
429:
401:
377:
374:
371:
351:
346:
342:
338:
333:
329:
325:
322:
319:
314:
310:
287:
283:
279:
276:
273:
268:
264:
260:
255:
251:
235:
232:
229:Shannon coding
227:Main article:
224:
221:
206:Shannon coding
176:
173:
157:Huffman coding
149:
148:
137:
117:
112:
108:
104:
99:
95:
91:
88:
85:
80:
76:
55:
29:Claude Shannon
27:, named after
15:
9:
6:
4:
3:
2:
4500:
4489:
4486:
4484:
4481:
4479:
4476:
4475:
4473:
4457:
4453:
4445:
4443:
4435:
4434:
4431:
4425:
4422:
4421:
4419:
4415:
4409:
4406:
4405:
4403:
4399:
4393:
4390:
4388:
4385:
4383:
4380:
4378:
4375:
4373:
4370:
4368:
4365:
4363:
4360:
4356:
4353:
4352:
4351:
4348:
4346:
4343:
4339:
4336:
4334:
4331:
4330:
4329:
4326:
4325:
4323:
4321:
4317:
4305:
4302:
4300:
4297:
4296:
4295:
4292:
4288:
4285:
4283:
4280:
4278:
4275:
4274:
4272:
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4260:
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4235:Video quality
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3902:Speech coding
3900:
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3897:Sound quality
3895:
3893:
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3875:
3873:
3872:Dynamic range
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3673:
3671:
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3655:
3652:
3651:
3649:
3644:
3642:
3639:
3638:
3637:LZ77 + Range
3636:
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3622:
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3043:
3036:
3030:
3023:
3017:
3009:
3008:
3001:
2994:
2988:
2986:
2984:
2968:on 2019-12-03
2964:
2960:
2956:
2952:
2948:
2941:
2934:
2930:
2922:
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2150:
2147:
2146:
2143:
2142:
2141:
2134:
2122:
2118:
2115:
2112:
2109:
2108:probabilities
2105:
2104:
2103:
2095:
2093:
2081:
2077:
2063:
2046:
2040:
2032:
2028:
2009:
2005:
2001:
1996:
1992:
1986:
1982:
1976:
1971:
1968:
1965:
1961:
1956:
1953:
1947:
1941:
1918:
1915:
1909:
1903:
1900:
1895:
1891:
1885:
1880:
1877:
1874:
1870:
1866:
1861:
1857:
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1823:
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1809:
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1759:
1753:
1748:
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1501:
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1459:
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1422:
1421:
1417:
1414:
1411:
1408:
1405:
1386:
1382:
1378:
1373:
1369:
1365:
1355:Word lengths
1354:
1353:
1349:
1346:
1343:
1340:
1337:
1335:...in binary
1334:
1333:
1329:
1326:
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1317:
1314:
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1303:
1300:
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1238:
1235:
1232:
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1226:
1207:
1203:
1199:
1194:
1190:
1186:
1176:Word lengths
1175:
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1164:
1161:
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1070:
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1056:Word lengths
1055:
1054:
1050:
1047:
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1038:
1022:
1018:
1014:
1009:
1005:
1001:
994:
993:
989:
986:
983:
980:
977:
974:
973:
969:
966:
963:
960:
957:
954:
953:
950:
949:
948:
929:
925:
921:
916:
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908:
902:
897:
893:
883:
869:
866:
860:
854:
847:
835:
832:
829:
826:
823:
820:
819:
815:
812:
809:
806:
803:
800:
799:
795:
792:
789:
786:
783:
780:
779:
776:
775:
774:
765:
749:
745:
736:
718:
714:
693:
671:
667:
663:
658:
654:
650:
645:
641:
637:
632:
628:
624:
619:
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611:
608:
605:
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572:
569:
566:
563:
553:
549:
543:
540:
537:
532:
529:
526:
522:
518:
513:
509:
504:
501:
498:
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489:
481:
480:
479:
463:
459:
455:
452:
449:
444:
440:
436:
431:
427:
417:
413:
399:
391:
372:
344:
340:
336:
331:
327:
323:
317:
312:
308:
285:
281:
277:
274:
271:
266:
262:
258:
253:
249:
239:
230:
220:
218:
214:
209:
207:
203:
197:
195:
189:
187:
180:
172:
170:
166:
161:
158:
154:
146:
141:
140:Fano's method
138:
135:
131:
110:
106:
102:
97:
93:
89:
83:
78:
74:
53:
45:
42:
41:
40:
38:
34:
30:
26:
22:
4408:Hutter Prize
4372:Quantization
4277:Compensation
4071:Quantization
3794:Compensation
3360:Shannon–Fano
3359:
3300:Entropy type
3235:
3229:
3216:
3185:
3179:
3163:
3143:
3136:
3127:
3121:. Retrieved
3107:
3099:
3094:
3086:
3081:
3073:
3068:
3060:
3055:
3047:
3042:
3034:
3029:
3021:
3016:
3006:
3000:
2992:
2970:. Retrieved
2963:the original
2950:
2946:
2933:
2920:
2811:
2760:
2756:
2685:
2636:
2571:
2452:
2444:
2329:
2218:
2214:
2210:
2139:
2101:
2082:
2078:
2074:
1933:
1671:
1568:
1560:
1445:
1266:
1126:
884:
843:
771:
587:
418:
414:
240:
237:
216:
212:
210:
205:
201:
199:
193:
190:
185:
182:
178:
162:
150:
139:
43:
24:
18:
4367:Prefix code
4220:Frame types
4041:Color space
3867:Convolution
3597:LZ77 + ANS
3508:Incremental
3481:Other types
3400:Levenshtein
3238:: 379–423.
3169:Huffman, D.
3115:APPNOTE.TXT
2579:independent
2092:file format
217:Fano coding
37:prefix code
33:Robert Fano
4472:Categories
4424:Mark Adler
4382:Redundancy
4299:Daubechies
4282:Estimation
4215:Frame rate
4137:Daubechies
4097:Chain code
4056:Macroblock
3862:Companding
3799:Estimation
3719:Daubechies
3425:Lempel–Ziv
3385:Exp-Golomb
3313:Arithmetic
3205:References
3123:2008-01-06
2972:3 December
2789:Codewords
2307:Codewords
1423:Codewords
1244:Codewords
208:by Yeung.
194:suboptimal
153:suboptimal
4401:Community
4225:Interlace
3611:Zstandard
3390:Fibonacci
3380:Universal
3338:Canonical
2959:212439287
2897:≈
2853:⋅
2835:⋅
2490:−
2469:≤
2421:≈
2389:⋅
2353:⋅
2002:
1962:∑
1957:−
1871:∑
1854:
1814:∑
1810:−
1785:
1772:−
1739:∑
1735:≤
1695:∑
1641:
1628:−
1625:≤
1622:⌉
1609:
1596:−
1593:⌈
1537:≈
1493:⋅
1469:⋅
1392:⌉
1379:
1366:−
1363:⌈
1213:⌉
1200:
1187:−
1184:⌈
1093:⌉
1080:
1067:−
1064:⌈
1015:
1002:−
935:⌉
922:
909:−
906:⌈
567:≥
541:−
523:∑
456:≥
453:⋯
450:≥
437:≥
376:⌉
370:⌈
350:⌉
337:
324:−
321:⌈
275:…
116:⌉
103:
90:−
87:⌈
4387:Symmetry
4355:Timeline
4338:FM-index
4183:Bit rate
4176:Concepts
4024:Concepts
3887:Sampling
3840:Bit rate
3833:Concepts
3535:Sequitur
3370:Tunstall
3343:Modified
3333:Adaptive
3291:Lossless
3171:(1952).
2953:(5): 5.
2512:, where
1350:0.11011
1347:0.10110
1344:0.10010
1341:0.01100
1338:0.00000
362:. Here,
4345:Entropy
4294:Wavelet
4273:Motion
4132:Wavelet
4112:Fractal
4107:Deflate
4090:Methods
3877:Latency
3790:Motion
3714:Wavelet
3631:LHA/LZH
3581:Deflate
3530:Re-Pair
3525:Grammar
3355:Shannon
3328:Huffman
3284:methods
2890:symbols
2769:Symbol
2694:Symbol
2414:symbols
2228:Symbol
2148:Symbol
2128:Example
2085:IMPLODE
2027:entropy
2025:is the
1530:symbols
1275:Symbol
1136:Symbol
955:Symbol
846:entropy
781:Symbol
768:Example
388:is the
4456:codecs
4417:People
4320:Theory
4287:Vector
3804:Vector
3621:Brotli
3571:Hybrid
3470:Snappy
3323:Golomb
3151:
3129:trees.
2957:
2749:0.128
2746:0.154
2743:0.154
2740:0.179
2737:0.385
2714:Count
2263:0.128
2260:0.154
2257:0.154
2254:0.179
2251:0.385
2203:0.128
2200:0.154
2197:0.154
2194:0.179
2191:0.385
2168:Count
2029:, and
1934:Here,
1330:0.872
1327:0.718
1324:0.564
1321:0.385
1318:0.000
1310:0.128
1307:0.154
1304:0.154
1301:0.179
1298:0.385
1171:0.128
1168:0.154
1165:0.154
1162:0.179
1159:0.385
1051:2.963
1048:2.700
1045:2.700
1042:2.480
1039:1.379
990:0.128
987:0.154
984:0.154
981:0.179
978:0.385
882:bits.
836:0.128
833:0.154
830:0.154
827:0.179
824:0.385
801:Count
175:Naming
4247:parts
4245:Codec
4210:Frame
4168:Video
4152:SPIHT
4061:Pixel
4016:Image
3970:ACELP
3941:ADPCM
3931:μ-law
3926:A-law
3919:parts
3917:Codec
3825:Audio
3764:ACELP
3752:ADPCM
3729:SPIHT
3670:Lossy
3654:bzip2
3645:LZHAM
3601:LZFSE
3503:Delta
3395:Gamma
3375:Unary
3350:Range
3176:(PDF)
2966:(PDF)
2955:S2CID
2943:(PDF)
2925:Notes
870:2.186
4259:DPCM
4066:PSNR
3997:MDCT
3990:WLPC
3975:CELP
3936:DPCM
3784:WLPC
3769:CELP
3747:DPCM
3697:MDCT
3641:LZMA
3542:LDCT
3520:DPCM
3465:LZWL
3455:LZSS
3450:LZRW
3440:LZJB
3149:ISBN
2974:2019
2900:2.23
2849:bits
2804:111
2801:110
2798:101
2795:100
2424:2.28
2385:bits
2349:bits
2322:111
2319:110
1540:2.62
1489:bits
1465:bits
1438:110
1435:101
1432:100
1429:011
1259:101
1256:100
1253:011
1250:010
192:but
31:and
4304:DWT
4254:DCT
4198:VBR
4193:CBR
4188:ABR
4147:EZW
4142:DWT
4127:RLE
4117:KLT
4102:DCT
3985:LSP
3980:LAR
3965:LPC
3958:FFT
3855:VBR
3850:CBR
3845:ABR
3779:LSP
3774:LAR
3759:LPC
3724:DWT
3709:FFT
3704:DST
3692:DCT
3591:LZS
3586:LZX
3562:RLE
3557:PPM
3552:PAQ
3547:MTF
3515:DMC
3493:CTW
3488:BWT
3460:LZW
3445:LZO
3435:LZ4
3430:842
3240:doi
3190:doi
2831:bit
2717:15
2535:min
2525:min
2498:min
2316:10
2313:01
2310:00
2171:15
2090:ZIP
1993:log
1845:log
1776:log
1632:log
1600:log
1426:00
1370:log
1247:00
1191:log
1071:log
1006:log
913:log
804:15
737:of
588:so
328:log
94:log
4474::
4122:LP
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