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567:, where index is the index to a dictionary entry representing a previously seen sequence, and token is the next token from the input that makes this entry unique in the dictionary. Note how the algorithm is greedy, and so nothing is added to the table until a unique making token is found. The algorithm is to initialize last matching index = 0 and next available index = 1 and then, for each token of the input stream, the dictionary searched for a match:
694:-structured dictionary is full, a simple re-use/recovery algorithm is used to ensure that the dictionary can keep adapting to changing data. A counter cycles through the dictionary. When a new entry is needed, the counter steps through the dictionary until a leaf node is found (a node with no dependents). This is deleted and the space re-used for the new entry. This is simpler to implement than LRU or LFU and achieves equivalent performance.
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characters from that position into the current position". How can ten characters be copied over when only four of them are actually in the buffer? Tackling one byte at a time, there is no problem serving this request, because as a byte is copied over, it may be fed again as input to the copy command.
562:
The LZ78 algorithms compress sequential data by building a dictionary of token sequences from the input, and then replacing the second and subsequent occurrence of the sequence in the data stream with a reference to the dictionary entry. The observation is that the number of repeated sequences is a
530:
In the PalmDoc format, a length–distance pair is always encoded by a two-byte sequence. Of the 16 bits that make up these two bytes, 11 bits go to encoding the distance, 3 go to encoding the length, and the remaining two are used to make sure the decoder can identify the first byte as the beginning
553:. Literals, lengths, and a symbol to indicate the end of the current block of data are all placed together into one alphabet. Distances can be safely placed into a separate alphabet; because a distance only occurs just after a length, it cannot be mistaken for another kind of symbol or vice versa.
511:
Even though all LZ77 algorithms work by definition on the same basic principle, they can vary widely in how they encode their compressed data to vary the numerical ranges of a length–distance pair, alter the number of bits consumed for a length–distance pair, and distinguish their length–distance
454:
Considering the above, especially if the compression of data runs is expected to predominate, the window search should begin at the end of the window and proceed backwards, since run patterns, if they exist, will be found first and allow the search to terminate, absolutely if the current maximal
130:
that can be achieved. It is then shown that there exists finite lossless encoders for every sequence that achieve this bound as the length of the sequence grows to infinity. In this sense an algorithm based on this scheme produces asymptotically optimal encodings. This result can be proven more
102:
Since LZ77 encodes and decodes from a sliding window over previously seen characters, decompression must always start at the beginning of the input. Conceptually, LZ78 decompression could allow random access to the input if the entire dictionary were known in advance. However, in practice the
520:
The algorithm illustrated in Lempel and Ziv's original 1977 article outputs all its data three values at a time: the length and distance of the longest match found in the buffer, and the literal that followed that match. If two successive characters in the input stream could be encoded only as
351:
of the copy-from position. The operation is thus equivalent to the statement "copy the data you were given and repetitively paste it until it fits". As this type of pair repeats a single copy of data multiple times, it can be used to incorporate a flexible and easy form of
335:. The encoder needs to keep this data to look for matches, and the decoder needs to keep this data to interpret the matches the encoder refers to. The larger the sliding window is, the longer back the encoder may search for creating references.
374:. Then as the search pointer proceeds past the search window and forward, as far as the run pattern repeats in the input, the search and input pointers will be in sync and match characters until the run pattern is interrupted. Then
671:
is that when a match is not found, the current input stream character is assumed to be the first character of an existing string in the dictionary (since the dictionary is initialized with all possible characters), so only the
299:
LZ77 algorithms achieve compression by replacing repeated occurrences of data with references to a single copy of that data existing earlier in the uncompressed data stream. A match is encoded by a pair of numbers called a
563:
good measure of the non random nature of a sequence. The algorithms represent the dictionary as an n-ary tree where n is the number of tokens used to form token sequences. Each dictionary entry is of the form
256:
571:. If a match is found, then last matching index is set to the index of the matching entry, nothing is output, and last matching index is left representing the input so far. Input is processed until a match is
538:, the size in bytes of a length–distance pair can be specified inside the first byte of the length–distance pair itself; depending on whether the first byte begins with a 0, 10, 110, or 111 (when read in
601:
is longer than the original input but compression ratio improves considerably as the dictionary grows, and in binary the indexes need not be represented by any more than the minimum number of bits.
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characters are read. But mirroring the encoding process, since the pattern is repetitive, the read pointer need only trail in sync with the write pointer by a fixed distance equal to the run length
359:
Another way to see things is as follows: While encoding, for the search pointer to continue finding matched pairs past the end of the search window, all characters from the first match at offset
455:
matching sequence length is met, or judiciously, if a sufficient length is met, and finally for the simple possibility that the data is more recent and may correlate better with the next input.
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579:, and the algorithm outputs last matching index, followed by token, then resets last matching index = 0 and increments next available index. As an example consider the sequence of tokens
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characters are read to the output, this corresponds to a single run unit appended to the output buffer. At this point, the read pointer could be thought of as only needing to return int(
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improves on LZ77 by using a 1-bit flag to indicate whether the next chunk of data is a literal or a length–distance pair, and using literals if a length–distance pair would be longer.
338:
It is not only acceptable but frequently useful to allow length-distance pairs to specify a length that actually exceeds the distance. As a copy command, this is puzzling: "Go back
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is an LZ78-based algorithm that uses a dictionary pre-initialized with all possible characters (symbols) or emulation of a pre-initialized dictionary. The main improvement of
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and forward to the end of the search window must have matched input, and these are the (previously seen) characters that comprise a single run unit of length
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In the second of the two papers that introduced these algorithms they are analyzed as encoders defined by finite-state machines. A measure analogous to
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is an LZ78-based algorithm that was developed for use in real-time communications systems (originally modems) and standardized by CCITT/ITU as
593:. Note that the last A is not represented yet as the algorithm cannot know what comes next. In practice an EOF marker is added to the input –
628:. The token "B" is output, preceded by the sequence represented by dictionary entry 1. Entry 1 is an 'A' (followed by "entry 0" – nothing) so
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is output (which may be the pre-initialized dictionary index corresponding to the previous (or the initial) input character). Refer to the
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and others. Besides their academic influence, these algorithms formed the basis of several ubiquitous compression schemes, including
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When the copy-from position makes it to the initial destination position, it is consequently fed data that was pasted from the
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d := distance to start of match l := length of match c := char following match in input
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is developed for individual sequences (as opposed to probabilistic ensembles). This measure gives a bound on the
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To spot matches, the encoder must keep track of some amount of the most recent data, such as the last 2
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constructed by LZ78—however, they are only equivalent when the entire data is intended to be decompressed.
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dictionary is created during encoding and decoding by creating a new phrase whenever a token is output.
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Decompression consists of rebuilding the dictionary from the compressed sequence. From the sequence
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1128:. Faculty of Electrical Engineering and Computing, University of Zagreb. 1997. Archived from
1102:. Faculty of Electrical Engineering and Computing, University of Zagreb. 1997. Archived from
516:(raw data encoded as itself, rather than as part of a length–distance pair). A few examples:
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The following pseudocode is a reproduction of the LZ77 compression algorithm sliding window.
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bit orientation), the length of the entire length–distance pair can be 1 to 4 bytes.
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785:(September 1978). "Compression of Individual Sequences via Variable-Rate Coding".
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respectively. These two algorithms form the basis for many variations including
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match := longest repeated occurrence of input that begins in window
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d := 0 l := 0 c := first char of input
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Faculty of
Electrical Engineering and Computing, University of Zagreb
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731:(May 1977). "A Universal Algorithm for Sequential Data Compression".
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Adaptive data compression system with systolic string matching logic
251:{\displaystyle \limsup _{n}{\frac {1}{n}}l_{LZ78}(X_{1:n})\leq h(X)}
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886:"IEEE Medal of Honor Goes to Data Compression Pioneer Jacob Ziv"
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429:≠0) times to the start of that single buffered run unit, read
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As of 2008, the most popular LZ77-based compression method is
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literals, the length of the length–distance pair would be 0.
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https://math.mit.edu/~goemans/18310S15/lempel-ziv-notes.pdf
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is added to the output. The second pair from the input is
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Similar theorems apply to other versions of LZ algorithm.
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IEEE Spectrum: Technology, Engineering, and
Science News
855:"Milestones:Lempel–Ziv Data Compression Algorithm, 1977"
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and the output sequence of the compressed data would be
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is a binary source that is stationary and ergodic, then
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characters behind it in the uncompressed stream". (The
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for example. Note also that in this case the output
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950:(2nd ed.). Hoboken, N.J: Wiley-Interscience.
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863:Institute of Electrical and Electronics Engineers
624:and results in entry number 2 in the dictionary,
496:+ 1 chars from front of window s := pop
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451:characters have been copied to output in total.
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575:found. Then a new dictionary entry is created,
636:is added to the dictionary as the next entry,
308:characters is equal to the characters exactly
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534:In the implementation used for many games by
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612:, and the first from the sequence would be
110:in 2004. In 2021 Jacob Ziv was awarded the
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114:for his involvement in their development.
998:"An Explanation of the Deflate Algorithm"
946:Cover, Thomas M.; Thomas, Joy A. (2006).
800:
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608:the first entry is always the terminator
577:dictionary = {last matching index, token}
117:
52:in 1977 and 1978. They are also known as
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331:, which is why LZ77 is sometimes called
788:IEEE Transactions on Information Theory
734:IEEE Transactions on Information Theory
378:characters have been matched in total,
14:
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583:which would assemble the dictionary;
131:directly, as for example in notes by
996:Feldspar, Antaeus (23 August 1997).
680:article for implementation details.
656:removing the spaces and EOF marker.
27:Lossless data compression algorithms
287:is the entropy rate of the source.
24:
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837:"Lossless Data Compression: LZ78"
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2377:Lossless compression algorithms
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586:0 {0,_} 1 {0,A} 2 {1,B} 3 {0,B}
964:
948:Elements of information theory
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632:is added to the output. Next
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143:LZ78 is universal and entropic
138:Formally, (Theorem 13.5.2 ).
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106:The algorithms were named an
569:{last matching index, token}
531:of such a two-byte sequence.
91:They are both theoretically
7:
972:"QFS Compression (RefPack)"
859:IEEE Global History Network
697:
565:dictionary = {index, token}
10:
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2222:Compressed data structures
1544:RLE + BWT + MTF + Huffman
1212:Asymmetric numeral systems
333:sliding-window compression
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1581:Discrete cosine transform
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1511:LZ77 + Huffman + context
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258:with probability 1. Here
39:lossless data compression
2286:Smallest grammar problem
811:10.1109/TIT.1978.1055934
757:10.1109/TIT.1977.1055714
549:; it combines LZSS with
316:is sometimes called the
2227:Compressed suffix array
1776:Nyquist–Shannon theorem
648:is output resulting in
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389:Upon decoding , again,
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44:published in papers by
1076:. 1997. Archived from
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492:(d, l, c) discard
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128:data compression ratio
118:Theoretical efficiency
2256:Kolmogorov complexity
2124:Video characteristics
1501:LZ77 + Huffman + ANS
824:US Patent No. 5532693
282:
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163:
2346:Compression software
1940:Compression artifact
1896:Psychoacoustic model
1096:"The LZ78 algorithm"
1066:"The LZ77 algorithm"
1061:at Wikimedia Commons
1049:at Wikimedia Commons
386:, and the code is .
342:characters and copy
302:length-distance pair
262:
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2336:Compression formats
1975:Texture compression
1970:Standard test image
1786:Silence compression
1122:"The LZW algorithm"
913:(14 October 2005).
674:last matching index
469:input is not empty
370:, which must equal
354:run-length encoding
146: —
124:information entropy
112:IEEE Medal of Honor
97:explicit dictionary
2382:Israeli inventions
2244:Information theory
2099:Display resolution
1925:Chroma subsampling
1314:Byte pair encoding
1259:Shannon–Fano–Elias
915:"Lempel–Ziv notes"
884:Joanna, Goodrich.
400:. When the first
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80:algorithm used in
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1132:on 7 January 2013
1106:on 7 January 2013
1080:on 7 January 2013
1057:Media related to
1045:Media related to
1002:comp.compression
957:978-0-471-24195-9
280:{\textstyle h(X)}
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16:(Redirected from
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1047:LZ77 algorithm
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1035:External links
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938:
927:on 28 May 2021
902:
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865:. 22 July 2014
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2302:Hutter Prize
2266:Quantization
2171:Compensation
1965:Quantization
1688:Compensation
1318:
1254:Shannon–Fano
1194:Entropy type
1134:. Retrieved
1130:the original
1125:
1108:. Retrieved
1104:the original
1099:
1082:. Retrieved
1078:the original
1069:
1021:
1009:. Retrieved
1001:
991:
979:. Retrieved
975:
966:
947:
941:
929:. Retrieved
922:the original
905:
893:. Retrieved
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879:
867:. Retrieved
858:
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819:
792:
786:
773:
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105:
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90:
57:
53:
37:are the two
34:
30:
29:
2261:Prefix code
2114:Frame types
1935:Color space
1761:Convolution
1491:LZ77 + ANS
1402:Incremental
1375:Other types
1294:Levenshtein
976:Niotso Wiki
690:. When the
512:pairs from
2366:Categories
2318:Mark Adler
2276:Redundancy
2193:Daubechies
2176:Estimation
2109:Frame rate
2031:Daubechies
1991:Chain code
1950:Macroblock
1756:Companding
1693:Estimation
1613:Daubechies
1319:Lempel–Ziv
1279:Exp-Golomb
1207:Arithmetic
1011:9 November
1007:. zlib.net
981:9 November
931:9 November
911:Peter Shor
895:18 January
869:9 November
779:Ziv, Jacob
725:Ziv, Jacob
711:References
650:A AB B A$
540:big-endian
459:Pseudocode
418:) + (1 if
320:instead.)
133:Peter Shor
42:algorithms
18:Lempel–Ziv
2295:Community
2119:Interlace
1505:Zstandard
1284:Fibonacci
1274:Universal
1232:Canonical
1004:newsgroup
797:CiteSeerX
743:CiteSeerX
606:0A1B0B1$
599:0A1B0B1$
349:beginning
234:≤
50:Jacob Ziv
2281:Symmetry
2249:Timeline
2232:FM-index
2077:Bit rate
2070:Concepts
1918:Concepts
1781:Sampling
1734:Bit rate
1727:Concepts
1429:Sequitur
1264:Tunstall
1237:Modified
1227:Adaptive
1185:Lossless
698:See also
638:3 {0,B}
614:1 {0,A}
610:0 {...}
514:literals
314:distance
310:distance
76:and the
2239:Entropy
2188:Wavelet
2167:Motion
2026:Wavelet
2006:Fractal
2001:Deflate
1984:Methods
1771:Latency
1684:Motion
1608:Wavelet
1525:LHA/LZH
1475:Deflate
1424:Re-Pair
1419:Grammar
1249:Shannon
1222:Huffman
1178:methods
1136:22 June
1110:22 June
1084:22 June
765:9267632
688:V.42bis
595:AABBA$
547:DEFLATE
78:DEFLATE
2350:codecs
2311:People
2214:Theory
2181:Vector
1698:Vector
1515:Brotli
1465:Hybrid
1364:Snappy
1217:Golomb
954:
799:
763:
745:
616:. The
591:0A1B0B
502:repeat
490:output
487:end if
447:until
318:offset
306:length
2141:parts
2139:Codec
2104:Frame
2062:Video
2046:SPIHT
1955:Pixel
1910:Image
1864:ACELP
1835:ADPCM
1825:ÎĽ-law
1820:A-law
1813:parts
1811:Codec
1719:Audio
1658:ACELP
1646:ADPCM
1623:SPIHT
1564:Lossy
1548:bzip2
1539:LZHAM
1495:LZFSE
1397:Delta
1289:Gamma
1269:Unary
1244:Range
925:(PDF)
918:(PDF)
761:S2CID
706:(LZS)
654:AABBA
626:{1,B}
581:AABBA
467:while
382:>
2153:DPCM
1960:PSNR
1891:MDCT
1884:WLPC
1869:CELP
1830:DPCM
1678:WLPC
1663:CELP
1641:DPCM
1591:MDCT
1535:LZMA
1436:LDCT
1414:DPCM
1359:LZWL
1349:LZSS
1344:LZRW
1334:LZJB
1138:2012
1112:2012
1086:2012
1013:2014
983:2014
952:ISBN
933:2014
897:2021
871:2014
692:trie
684:BTLZ
558:LZ78
525:LZSS
483:else
479:then
422:mod
340:four
295:LZ77
84:and
70:LZMA
66:LZSS
56:and
48:and
35:LZ78
33:and
31:LZ77
2198:DWT
2148:DCT
2092:VBR
2087:CBR
2082:ABR
2041:EZW
2036:DWT
2021:RLE
2011:KLT
1996:DCT
1879:LSP
1874:LAR
1859:LPC
1852:FFT
1749:VBR
1744:CBR
1739:ABR
1673:LSP
1668:LAR
1653:LPC
1618:DWT
1603:FFT
1598:DST
1586:DCT
1485:LZS
1480:LZX
1456:RLE
1451:PPM
1446:PAQ
1441:MTF
1409:DMC
1387:CTW
1382:BWT
1354:LZW
1339:LZO
1329:LZ4
1324:842
807:doi
753:doi
678:LZW
669:LZW
665:LZW
660:LZW
652:or
646:A$
642:1$
573:not
344:ten
148:If
86:ZIP
82:PNG
74:GIF
62:LZW
58:LZ2
54:LZ1
2368::
2016:LP
1847:FT
1840:DM
1392:CM
1124:.
1098:.
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1068:.
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759:.
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727:;
634:0B
630:AB
622:1B
475:if
471:do
393:=
356:.
325:KB
207:78
135:.
88:.
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2348:(
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813:.
809::
767:.
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618:A
498:l
494:l
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445:R
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427:R
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420:L
416:R
413:L
411:/
409:L
405:R
402:L
398:R
395:L
391:D
384:D
380:L
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372:D
368:R
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361:D
275:)
272:X
269:(
266:h
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243:X
240:(
237:h
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226:n
223::
220:1
216:X
212:(
204:Z
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191:n
188:1
181:n
156:X
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.