Binary Golomb Codes. CSE 589 Applied Algorithms Autumn Constructing a Binary Golomb Code. Example. Example. Comparison of GC with Entropy

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1 CSE 589 pplied lgorithms utumn Golom Coding rithmetic Coding LZW Sequitur Binry Golom Codes Binry source with s much more frequent thn s. Vrile-to-vrile length code Prefix code. Golom code of order 4 input output 2 Exmple Constructing Binry Golom Code input output Let p e the proility of Choose the order k such tht lest likely p k + k p < p k most likely k- 3 4 p =.89 p 7 = p 6 =.53 p 5 =.558 Order = 6 Exmple input output Comprison of GC with Entropy order p GC entropy entropy 5 6

2 Notes on Binry Golom Code Very effective s inry run-length code s p goes to effectiveness ner entropy There re dptive Golom codes, so tht the order cn chnge ccording to oserved frequency of s. rithmetic Coding Huffmn coding works well for lrger lphets nd gets to within one it of the entropy lower ound. Cn we do etter. Yes Bsic ide in rithmetic coding: represent ech string x of length n y n intervl [l,r) in [,). We ssume n is known. The width r-l of the intervl [l,r) represents the proility of x occurring. The intervl [l,r) cn itself e represented y ny numer, clled tg, within the hlf open intervl. The k significnt its of the tg.t t 2 t 3... is the code of x. Tht is,..t t 2 t 3...t k... is in the intervl [l,r). 7 8 Exmple of rithmetic Coding () Some Tgs re Better thn Others /3. tg must e in the hlf open intervl. 2. tg cn e chosen to e (l+r)/2. 3. code is significnt its of the tg. /3 2/3 5/ /27... tg = 7/27 =... code = 2/3 / /27... Using tg = (l+r)/2 tg = 3/27 =... code = lterntive tg = 4/37 =... code = 9 P() = /3, P() = 2/3. Exmple of Codes /27 /27 3/27 5/ /27... tg = (l+r)/2 code /27... / / / its/symol.92 entropy lower ound Code Genertion from Tg If inry tg is.t t 2 t 3... = (r+l)/2 in [l,r) then we wnt to choose k to form the code t t 2...t k. Short code: choose k to e s smll s possile so tht l <.t t 2...t k... < r. Good if n is known. Gurnteed code: choose k = log 2 ( r l) + l <.t t 2...t k < r for ny its for fixed length strings provides good prefix code. exmple: [...,...), tg =... Short code: Gurnteed code: 2 2

3 Gurnteed Code Exmple rithmetic Coding lgorithm P() = /3, P() = 2/3. /27 /27 3/27 tg = (l+r)/2 short code Prefix code / /27 / / /27... P( ), P( 2 ),, P( m ) C( i ) = P( ) + P( 2 ) + + P( i- ) Encode x x 2...x n Initilize l := nd r := ; for i = to n do w := r - l; l := l + w * C(x i ); r := l + w * P(x i ); t := (l+r)/2; choose code for the tg 27/ rithmetic Coding Exmple rithmetic Coding Exercise P() = /4, P() = /2, P(c) = /4 C() =, C() = /4, C(c) = 3/4 c w := r - l; l := l + w C(x); r := l + w P(x) symol w l r /4 /4 /6 3/6 c /8 5/32 6/32 /32 5/32 2/28 P() = /4, P() = /2, P(c) = /4 C() =, C() = /4, C(c) = 3/4 w := r - l; l := l + w C(x); r := l + w P(x) symol w l r tg = (5/32 + 2/28)/2 = 4/256 =... l =... r =... code = prefix code = tg = l = r = code = prefix code = 5 6 Decoding () Decoding (2) ssume the length is known to e 3. which converts to the tg... ssume the length is known to e 3. which converts to the tg output... output 7 8 3

4 Decoding (3) rithmetic Decoding lgorithm ssume the length is known to e 3. which converts to the tg output P( ), P( 2 ),, P( m ) C( i ) = P( ) + P( 2 ) + + P( i- ) Decode 2... m, numer of symols is n. Initilize l := nd r := ; t := m... for i = to n do w := r - l; find j such tht l + w * C( j ) < t < l + w * (C( j )+P( j )) output j ; l := l + w * C( j ); r := l + w * P( j ); 9 2 Decoding Exmple Prcticl rithmetic Coding P() = /4, P() = /2, P(c) = /4 C() =, C() = /4, C(c) = 3/4 tg =... = 5/32 w l r output /4 /4 /6 3/6 /8 5/32 6/32 c /32 5/32 2/28 Scling: By scling we cn keep l nd r in resonle rnge of vlues so tht w = r - l does not underflow. The code cn e produced progressively, not t the end. Complictes decoding some. Integer rithmetic coding voids floting point ltogether Notes on rithmetic Coding rithmetic codes come close to the entropy lower ound. Grouping symols is effective for rithmetic coding. rithmetic codes cn e used effectively on smll symol sets. dvntge over Huffmn. Context cn e dded so tht more thn one proility distriution cn e used. The est coders in the world use this method. There re very effective dptive rithmetic coding methods. Coding Does not use sttisticl knowledge of dt. Encoder: s the input is processed develop dictionry nd trnsmit the index of strings found in the dictionry. Decoder: s the code is processed reconstruct the dictionry to invert the process of encoding. Exmples: LZW, LZ77, Unix Compress, gzip, GIF

5 LZW Encoding lgorithm LZW Encoding Exmple () Repet find the longest mtch w in the dictionry output the index of w put w in the dictionry where ws the unmtched symol LZW Encoding Exmple (2) LZW Encoding Exmple (3) LZW Encoding Exmple (4) LZW Encoding Exmple (5)

6 LZW Encoding Exmple (6) LZW Decoding lgorithm Emulte the encoder in uilding the dictionry. Decoder is slightly ehind the encoder. initilize dictionry; decode first index to w; put w? in dictionry; repet decode the first symol s of the index; complete the previous dictionry entry with s; finish decoding the reminder of the index; put w? in the dictionry where w ws just decoded; 3 32 LZW Decoding Exmple () LZW Decoding Exmple (2) 2? LZW Decoding Exmple (2) LZW Decoding Exmple (3) 3?

7 LZW Decoding Exmple (3) LZW Decoding Exmple (4) 3 4? LZW Decoding Exmple (4) LZW Decoding Exmple (5) 3 4 5? LZW Decoding Exmple (5) LZW Decoding Exmple (6) ?

8 LZW Decoding Exmple (6) ? Trie Dt Structure for Encoder s Fredkin (96) 9 c d 2 c d 3 d r 4 r 3 r 5 4 r 6 r 7 r 8 c c d r 5 c 8 d r 6 9 r Encoder Uses Trie () Encoder Uses Trie (2) c d r c d r 5 c 8 d r c 8 d r r 2 3 r r c d r r c d r r c d r r c d r Decoder s Dt Structure Notes on Coding Simply n rry of strings 9 c d 2 c d 3 d r 4 r 3 r 5 4 r? 6 r 7 r 8 c r c d r r Extremely effective when there re repeted ptterns in the dt tht re widely spred. Where locl context is not s significnt. text some grphics progrm sources or inries Vrints of LZW re pervsive

9 Sequitur Nevill-Mnning nd Witten, 996. Uses context-free grmmr (without recursion) to represent string. The grmmr is inferred from the string. If there is structure nd repetition in the string then the grmmr my e very smll compred to the originl string. Clever encoding of the grmmr yields impressive compression rtios. Compression plus structure! Context-Free Grmmrs Invented y Chomsky in 959 to explin the grmmr of nturl lnguges. lso invented y Bckus in 959 to generte nd prse Fortrn. Exmple: terminls:, e nonterminls: S, Production Rules: S -> S, S ->, -> Se, S is the strt symol 49 5 Context-Free Grmmr Exmple S -> S S -> -> Se Exmple: nd e mtched s prentheses derivtion of eee S Se Se e ee eee hierrchicl prse tree S S e S e e S -> S + T S -> T T -> T*F T -> F F -> F -> (S) rithmetic Expressions derivtion of * ( + ) + S S+T T+T T*F+T F*F+T *F+T *(S)+F *(S+F)+T *(T+F)+T *(F+F)+T *(+F)+T *(+)+T *(+)+F *(+)+ T T * F S S + T F ( S ) S + T T F prse tree F F 5 52 Sequitur Principles Sequitur Exmple () Digrm Uniqueness: no pir of djcent symols (digrm) ppers more thn once in the grmmr. Rule Utility: Every production rule is used more thn once. These two principles re mintined s n invrint while inferring grmmr for the input string. S ->

10 Sequitur Exmple (2) Sequitur Exmple (3) S -> S -> e Sequitur Exmple (4) Sequitur Exmple (5) S -> e S -> ee Enforce digrm uniqueness. e occurs twice. Crete new rule ->e Sequitur Exmple (6) Sequitur Exmple (7) S -> S -> e 59 6

11 Sequitur Exmple (8) Sequitur Exmple (9) S -> e S -> ee Enforce digrm uniqueness. e occurs twice. Use existing rule Sequitur Exmple () Sequitur Exmple () S -> e S -> e Sequitur Exmple (2) Sequitur Exmple (3) S -> ee Enforce digrm uniqueness. e occurs twice. Use existing rule. S -> e Enforce digrm uniqueness occurs twice. Crete new rule

12 Sequitur Exmple (4) Sequitur Exmple (5) S -> BeB S -> BeB Sequitur Exmple (6) Sequitur Exmple (7) S -> BeB S -> BeBe Enforce digrm uniqueness. e occurs twice. Use existing rule ->e Sequitur Exmple (8) Sequitur Exmple (9) S -> BeB S -> BeB

13 Sequitur Exmple (2) Sequitur Exmple (2) S -> BeBe Enforce digrm uniqueness. e occurs twice. Use existing rule. S -> BeB Enforce digrm uniqueness. occurs twice. Use existing rule Sequitur Exmple (22) Sequitur Exmple (23) S -> BeBB Enforce digrm uniqueness. B occurs twice. Crete new rule C -> B. S -> CeBC C -> B Sequitur Exmple (24) Sequitur Exmple (25) S -> CeBCe C -> B Enforce digrm uniqueness. Ce occurs twice. Crete new rule D -> Ce. S -> DBD C -> B D -> Ce Enforce rule utility. C occurs only once. Remove C -> B

14 Sequitur Exmple (26) S -> DBD D -> Be The Hierrchy S -> DBD D -> Be B e S D B D B e e e e e e e Is there compression? In this smll exmple, proly not Sequitur lgorithm Input the first symol s to crete the production S -> s; repet mtch n existing rule: ->.XY ->.B. B -> XY B -> XY crete new rule: ->.XY. ->.C... B ->...XY... B ->...C... remove rule: C -> XY ->.B. B -> X X 2 X ->. X X 2 X k. k input new symol: S -> X X 2 X k until no symols left S -> X X 2 X k s Complexity The numer of non-input sequitur opertions pplied < 2n where n is the input length. mortized Complexity rgument Let s = the sum of the right hnd sides of ll the production rules. Let r = the numer of rules. We evlute 2s - r. Initilly 2s - r = ecuse s = nd r =. 2s - r > t ll times ecuse ech rule hs t lest symol on the right hnd side. 2s - r increses y 2 for every input opertion. 2s - r decreses y t lest for ech non-input sequitur rule pplied Sequitur Rule Complexity Digrm Uniqueness - mtch n existing rule. ->.XY B -> XY Digrm Uniqueness - crete new rule. ->.XY. B ->...XY... ->.B. B -> XY ->.C... B ->...C... C -> XY Rule Utility - Remove rule. s r - s r 2s - r -2 2s - r - Liner Time lgorithm There is dt structure to implement ll the sequitur opertions in constnt time. Production rules in n rry of douly linked lists. Ech production rule hs reference count of the numer of times used. Ech nonterminl points to its production rule. digrms stored in hsh tle for quick lookup. ->.B. B -> X X 2 X k s r ->. X X 2 X k s - r

15 S -> CeBCe C -> B Dt Structure Exmple S C e B C e 2 B 2 e C 2 B reference count current digrm digrm tle BC eb Ce e B 85 Grmmr Bsic Encoding Grmmr S -> DBD D -> Be Symol Code Grmmr Code e S B D # D B D # e # # B e 39 its Grmmr Code = ( s + r ) log 2 ( r + + ) r = numer of rules s = sum of right hnd sides = numer in originl symol lphet 86 Better Encoding of the Grmmr Nevill-Mnning nd Witten suggest more efficient encoding of the grmmr tht resemles LZ77. The first time nonterminl is sent, its right hnd side is trnsmitted insted. The second time nonterminl is sent the new production rule is estlished with pointer to the previous occurrence sent long with the length of the rule. Susequently, the nonterminl is represented y the index of the production rule. Compression Qulity Neville-Mnning nd Witten 997 size compress gzip sequitur PPMC i ook geo oj pic progc Files from the Clgry Corpus Units in its per chrcter (8 its) compress - sed on LZW gzip - sed on LZ77 PPMC - dptive rithmetic coding with context Notes on Sequitur Very new nd different from the stndrds. Yields compression nd hierrchicl structure simultneously. With clever encoding is competitive with the est of the stndrds. Prcticl liner time encoding nd decoding. lterntives Off-line lgorithms (i) find the most frequent digrm, (ii) find the longest repeted sustring 89 5

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