4.8 Huffman Codes. Wordle. Encoding Text. Encoding Text. Prefix Codes. Encoding Text

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1 2/26/2 Word A word a word coag. A word contrctd ot of on of th ntrctor ar: 4.8 Hffan Cod word contrctd ng th java at at word.nt word a randozd grdy agorth to ov th ackng rob Encodng Txt Q. Gvn a txt that 32 ybo (26 dffrnt ttr, ac, and o nctaton charactr), how can w ncod th txt n bt? Q. So ybo (, t, a, o,, n) ar d far or oftn than othr. How can w th to rdc or ncodng? Encodng Txt Q. Gvn a txt that 32 ybo (26 dffrnt ttr, ac, and nctaton charactr), how can w ncod th txt n bt? A. W can ncod 2 5 dffrnt ybo ng a fxd ngth of 5 bt r ybo. Th cad fxd ngth ncodng. Q. How do w know whn th nxt ybo bgn? Ex. c(a) = What? c(b) = c() = 3 4 Encodng Txt Q. So ybo (, t, a, o,, n) ar d far or oftn than othr. How can w th to rdc or ncodng? A. Encod th charactr wth fwr bt, and th othr wth or bt. Q. How do w know whn th nxt ybo bgn? A. U a araton ybo (k th a n Mor), or ak r that thr no abgty by nrng that no cod a rfx of anothr on. Ex. c(a) = What? c(b) = c() = Prfx Cod Dfnton. A rfx cod for a t S a fncton c that a ach to and n ch a way that for x,y S, x y, c(x) not a rfx of c(y). Ex. c(a) = c() = c(k) = c() = c() = Q. What th anng of? 5 6

2 2/26/2 Fxd v. Varab ncodng a b c d f frqncy(x) fxd ncodng varab ncodng, charactr Fxd: 3, bt Varab? (*45+3*3+3*2+3*6+4*9+4*5)* = 224, bt 25% avng Ota Prfx Cod Dfnton. Th avrag bt r ttr: f x frqncy of ttr x S ahabt ABL ( c) = f c( x) W wod k to fnd a rfx cod that ha th owt ob avrag bt r ttr. x 8 Nod abd wth charactr Path to a nod rrnt t cod Ex. c(a) = c() = c(k) = c() = c() = k a Ex. c(a) = c() = c(k) = c() = c() = k a Q. What ca abot th tr of a rfx cod? Q. What ca abot th tr of a rfx cod? A. Ony th av ar abd. 9 ABL ( T ) = f x dth ( x) T dth T (x) th dth whr x occr n T Q. How can th rfx cod b ad or ffcnt? 2

3 2/26/2 Dfnton. A tr f f vry nod that not a af ha two chdrn. Ca. Th bnary tr corrondng to th ota rfx cod f. w Q. How can th rfx cod b ad or ffcnt? A. Chang ncodng of and to a hortr on. Th tr now f. v 3 4 Ota Prfx Cod: Fa Start Dfnton. A tr f f vry nod that not a af ha two chdrn. Ca. Th bnary tr corrondng to th ota rfx cod f. Pf. (by contradcton) So T bnary tr of ota rfx cod and not f. Th an thr a nod wth ony on chd v. w Ca : th root; dt and v a th root Grdy aroach. Crat tr to-down, t S nto two t S and S 2 wth (aot) qa frqnc. Rcrvy bd tr for S and S 2. [Shannon-Fano, 949] Exa: f a =.32, f =.25, f k =.2, f =.8, f =.5 Ca 2: not th root t w b th arnt of dt and ak v b a chd of w n ac of v In both ca th nbr of bt ndd to ncod any af n th btr of v dcrad. Th rt of th tr not affctd. Cary th nw tr T ha a ar ABL than T. Contradcton. a k a k Ota Prfx Cod Ota Prfx Cod Q. Whr n th tr of an ota rfx cod hod ow frqncy ttr b acd? Obrvaton. Lowt frqncy t hod b at th owt v n tr of ota rfx cod. Obrvaton. For or than two ttr, th owt v away contan at at two av. Obrvaton. Th ordr n whch t aar n a v do not attr. Ca. Thr an ota rfx cod wth tr T* whr th two owt-frqncy ttr ar agnd to av that ar bng n T*. 7 8

4 2/26/2 Ota Prfx Cod: Hffan Encodng Ca. Thr an ota rfx cod wth tr T* whr th two owt-frqncy ttr ar agnd to av that ar bng n T*. Hffan ncodng. [Hffan, 952] Crat tr botto-: v v Mak two av for two owt-frqncy ttr y and z. Rcrvy bd tr for th rt ng a tattr for yz. Ota Prfx Cod: Hffan Encodng Hffan(S) { f S =2 { rtrn tr wth root and 2 av { t y and z b owt-frqncy ttr n S rov y and z fro S nrt nw ttr ω n S wth f ω =f y +f z T = Hffan(S ) T = add two chdrn y and z to af ω fro T rtrn T Exa: f a =.32, f =.25, f k =.2, f =.8, f = Ota Prfx Cod: Hffan Encodng Ota Prfx Cod: Hffan Encodng Hffan(S) { f S =2 { rtrn tr wth root and 2 av { t y and z b owt-frqncy ttr n S rov y and z fro S nrt nw ttr ω n S wth f ω =f y +f z T = Hffan(S ) T = add two chdrn y and z to af ω fro T rtrn T Q. What th rnnng t? Hffan(S) { f S =2 { rtrn tr wth root and 2 av { t y and z b owt-frqncy ttr n S rov y and z fro S nrt nw ttr ω n S wth f ω =f y +f z T = Hffan(S ) T = add two chdrn y and z to af ω fro T rtrn T Q. What th rnnng t? A. S aroach: T(n) = T(n-) + O(n) o O(n 2 ) Ung rorty q for S: T(n) = T(n-) + O(og n) o O(n og n) 2 22 Hffan Encodng: Otaty Ca. Th Hffan cod for S achv th n ABL of any rfx cod. Th chang n ABL whn ovng fro T to T (y and z rovd, ω addd): ABL(T )=ABL(T)-f ω Hffan Encodng: Otaty Th chang n ABL whn ovng fro T to T (y and z rovd, ω addd): ABL(T )=ABL(T)-f ω ABL(T) = f x dth T (x) = f y dth T (y)+ f z dth T (z)+ f x dth T (x),x y,z = ( f y + f z ) ( + dth T (ω))+ f x dth T (x),x y,z = f ω ( + dth T (ω))+ f x dth T (x),x y,z = f ω + f x dth T ' (x) ' = f ω + ABL(T' ) 23 24

5 2/26/2 Hffan Encodng: Otaty Hffan Encodng: Otaty Ca. Th Hffan cod for S achv th n ABL of any rfx cod. Proof. (by ndcton ovr n= S ) Ca. Th Hffan cod for S achv th n ABL of any rfx cod. Proof. (by ndcton) Ba: For n=2 thr no hortr cod than root and two av. Indcton Hyoth: So Hffan tr T for S wth ω ntad of y and z ota. (IH) Indcton St: (by contradcton) So Hffan tr T for S not ota. So thr o tr Z ch that ABL(Z) < ABL(T). Thn thr ao a tr Z for whch av y and z xt that ar bng and hav th owt frqncy ( obrvaton). Lt Z b Z wth y and z dtd, and thr forr arnt abd ω. Sary, T drvd fro T n or agorth. W know that ABL(Z )=ABL(Z)-f ω, a w a ABL(T )=ABL(T)-f ω. Bt ao ABL(Z) < ABL(T), o ABL(Z ) < ABL(T ). Contradcton wth IH