phu . o . ng x t Sx, v´o . i S l`a ma trˆa . nd¯u . `o . ng ch´eo. Trong tru . `o . ng ho . . p n`ay m˜a ho´a tˆo ´ iu . ucˆa ` n khai triˆe ˙’ na ˙’ nh th`anh c´ac th`anh phˆa ` n Karhunen-Lo`eve v`a biˆe ˙’ udiˆe ˜ nmˆo ˜ i th`anh phˆa ` n v´o . i sai sˆo ´ b`ınh phu . o . ng trung b`ınh bˇa ` ng nhau. 6.4 N´en ba ˙’ oto`an thˆong tin Trong mˆo . tsˆo ´ ´u . ng du . ng ta cˆa ` nlu . utr˜u . d˜u . liˆe . u ´ıt tˆo ´ nk´emvˆe ` khˆong gian nhu . ng pha ˙’ i ba ˙’ o to`an thˆong tin. Chˇa ˙’ ng ha . n, hˆo ` so . vˆe ` ydu . o . . chayd˜u . liˆe . u kinh doanh khˆong thˆe ˙’ d¯ ˆe ˙’ sai la . c. Phˆa ` n n`ay tˆa . p trung v`ao nh˜u . ng phu . o . ng ph´ap ch´ınh n´en ba ˙’ o to`an thˆong tin. Tı ˙’ lˆe . n´en cu ˙’ anh˜u . ng phu . o . ng ph´ap n`ay thu . `o . ng trong khoa ˙’ ng t`u . 2d¯ˆe ´ n 10. Ho . nn˜u . ac´o thˆe ˙’ ´ap du . ng d¯ˆo ´ iv´o . ia ˙’ nh nhi . phˆan v`a a ˙’ nh gi´a tri . x´am. Nhu . d¯˜a chı ˙’ ra trong Phˆa ` n 6.2, c´ac k˜y thuˆa . tn´en ba ˙’ o to`an thˆong tin thu . `o . ng gˆo ` m hai thao t´ac tu . o . ng d¯ˆo ´ id¯ˆo . clˆa . p: (1) t`ım mˆo . tbiˆe ˙’ udiˆe ˜ n kh´ac cu ˙’ aa ˙’ nh d¯ˆe ˙’ gia ˙’ mdu . th `u . a trong quan hˆe . gi˜u . a c´ac pixel v`a (2) m˜a ho´a trˆen biˆe ˙’ udiˆe ˜ nm´o . id¯ˆe ˙’ khu . ˙’ du . th `u . a trong bˆo . m˜a. Hai bu . ´o . c n`ay tu . o . ng ´u . ng v´o . i c´ac ph´ep to´an biˆe ´ nd¯ˆo ˙’ iv`am˜aho´ak´yhiˆe . ucu ˙’ a mˆo h`ınh m˜a ho´a nguˆo ` n (H`ınh 6.3). 6.4.1 M˜a ho´a d¯ˆo . d`ai thay d¯ˆo ˙’ i Phu . o . ng ph´ap d¯o . n gia ˙’ n nhˆa ´ tn´enba ˙’ o to`an thˆong tin a ˙’ nh l`a chı ˙’ gia ˙’ mduth`u . a trong m˜a ho´a. Du . th `u . a trong m˜a ho´a thu . `o . ng xuˆa ´ thiˆe . no . ˙’ nh˜u . ng a ˙’ nh d¯u . o . . c m˜a ho´a theo m˜a tu . . nhiˆen (d¯ˆo . d`ai khˆong d¯ˆo ˙’ i). Nhu . d¯ ˜a c h ´u ´y trong Phˆa ` n 6.1.1, ta c´o thˆe ˙’ khu . ˙’ du . th `u . a trong m˜a ho´a sao cho d¯ˆo . d`ai trung b`ınh cu ˙’ a c´ac t`u . m˜a l`a nho ˙’ nhˆa ´ t. D - ˆe ˙’ thu . . chiˆe . nd¯u . o . . cd¯iˆe ` u n`ay ta cˆa ` n g´an c´ac t`u . m˜a c´o d¯ˆo . d`ai ngˇa ´ n cho c´ac m´u . c x´am thu . `o . ng xuyˆen xuˆa ´ thiˆe . n. Du . ´o . i d¯ˆay ta s˜e tr`ınh b`ay mˆo . tsˆo ´ k˜y thuˆa . ttˆo ´ iu . u v`a gˆa ` ntˆo ´ iu . ud¯ˆe ˙’ xˆay du . . ng nh˜u . ng bˆo . m˜a c´o t´ınh chˆa ´ t n`ay. C´ac k˜y thuˆa . t n`ay d¯˜a d¯u . o . . c ph´at biˆe ˙’ u trong ngˆon ng˜u . cu ˙’ al´y thuyˆe ´ t thˆong tin. Thu . . cvˆa . y, c´ac k´y hiˆe . u nguˆo ` n c´o thˆe ˙’ hoˇa . c l`a c´ac m´u . c x´am cu ˙’ aa ˙’ nh hoˇa . c l`a t´ın hiˆe . u ra cu ˙’ amˆo . t thao t´ac biˆe ´ nd¯ˆo ˙’ im´u . c x´am (hiˆe . u c´ac pixel, d¯ˆo . d`ai cha . y, vˆan vˆan). 171 M˜a Huffman Trong hˆa ` uhˆe ´ t c´ac tˆa . p tin a ˙’ nh, mˆo . t v`ai gi´a tri . x´am xuˆa ´ thiˆe . nthu . `o . ng xuyˆen ho . n c´ac gi´a tri . kh´ac. Bo . ˙’ ivˆa . yc´omˆo . tphu . o . ng ph´ap n`ao d¯ˆe ˙’ g´an t`u . m˜a c´o d¯ˆo . d`ai ngˇa ´ nho . ncho nh˜u . ng gi´a tri . thu . `o . ng xuˆa ´ thiˆe . n v`a d`ai ho . nchonh˜u . ng gi´a tri . ´ıt xuˆa ´ thiˆe . n? D - ˆay ch´ınh l`a d¯iˆe ` u m`a Shannon v`a Fano d¯˜a ngh˜ı khi ho . d¯ u . a ra thuˆa . t to´an n´en d¯ˆa ` u tiˆen v`ao nˇam 1950. Tuy nhiˆen thuˆa . t to´an la . i mang tˆen cu ˙’ a Huffman khi ˆong cˆong bˆo ´ mˆo . t thuˆa . t to´an chi tiˆe ´ t v`a tˆo ´ tho . n trˆen mˆo . t b`ai b´ao nˇam 1952. Khi m˜a ho´a c´ac k´y hiˆe . ucu ˙’ a nguˆo ` n thˆong tin, m˜a ho´a Huffman cho biˆe ´ tsˆo ´ nho ˙’ nhˆa ´ t c´o thˆe ˙’ cu ˙’ a c´ac k´y hiˆe . u m˜a ho´a trˆen k´yhiˆe . u nguˆo ` n. Theo d¯i . nh l´y m˜a ho´a khˆong nhiˆe ˜ ubˆo . m˜a l`a tˆo ´ iu . ud¯ˆo ´ iv´o . i gi´a tri . n cho tru . ´o . c v`a r`ang buˆo . cc´ack´yhiˆe . u nguˆo ` nd¯u . o . . c m˜a ho´a t`u . ng c´ai mˆo . t. Thuˆa . t to´an Huffman l`a mˆo . t thuˆa . t to´an n´en d˜u . liˆe . usu . ˙’ du . ng x´ac suˆa ´ t nhˇa ` m gia ˙’ m d¯ ˆo . du . th `u . ad¯ˆo . d`ai c´ac t`u . m˜a d¯u . o . . csu . ˙’ du . ng d¯ˆe ˙’ biˆe ˙’ udiˆe ˜ n c´ac k´y hiˆe . ucu ˙’ a nguˆo ` n thˆong tin. D - ˆay l`a mˆo . tbˆo . m˜a tˆo ´ iu . u trong tru . `o . ng ho . . ptˆa ´ tca ˙’ c´ac k´y hiˆe . u c´o x´ac suˆa ´ tl`al˜uy th `u . a nguyˆen cu ˙’ a 1 2 . M˜a Huffman d¯u . o . . c xˆay du . . ng qua hai giai d¯oa . nnhu . sau. Giai d¯oa . n 1: Thu go . n nguˆo ` n. 1. sˇa ´ pxˆe ´ pc´ack´yhiˆe . u theo th´u . tu . . gia ˙’ m x´ac suˆa ´ t xuˆa ´ thiˆe . n; 2. tˆo ˙’ ho . . pliˆen tiˆe ´ p hai k´yhiˆe . u c´o x´ac suˆa ´ t nho ˙’ nhˆa ´ td¯ˆe ˙’ ta . o th`anh mˆo . t“k´yhiˆe . u ho . . p” c´o x´ac suˆa ´ tbˇa ` ng tˆo ˙’ ng hai x´ac suˆa ´ t n`ay. N´oi c´ach kh´ac ta thu go . n nguˆo ` n. 3. lˇa . pla . i c´ac bu . ´o . cthugo . n trˆen cho d¯ˆe ´ n khi c`on hai k´y hiˆe . u. H`ınh 6.8 minh ho . a qu´a tr`ınh m˜a ho´a xˆay du . . ng c´ac t`u . m˜a nhi . phˆan (c´o thˆe ˙’ ´ap du . ng tu . o . ng tu . . v´o . i c´ac m˜a Huffman K-ray). Cˆo . tth´u . nhˆa ´ t v`a th´u . hai tu . o . ng ´u . ng c´ac k´yhiˆe . u v`a c´ac x´ac suˆa ´ t xuˆa ´ thiˆe . n (d¯u . o . . csˇa ´ p theo th ´u . tu . . gia ˙’ mdˆa ` n). Ph´ep r´ut go . n d¯ ˆa ` u tiˆen: hai x´ac suˆa ´ t nho ˙’ nhˆa ´ t (0.06 v`a 0.04) d¯u . o . . ctˆo ˙’ ho . . p th`anh mˆo . t“k´yhiˆe . uho . . p” v´o . i x´ac suˆa ´ t0.06+0.04 = 0.1. K´yhiˆe . uho . . p v`a x´ac suˆa ´ ttu . o . ng ´u . ng c`ung v´o . inh˜u . ng k´y hiˆe . u c`on la . i (trong tru . `o . ng ho . . p n`ay l`a a 1 ,a 2 ,a 4 ,a 6 )s˜ed¯u . o . . csˇa ´ pxˆe ´ p theo th´u . tu . . x´ac suˆa ´ t gia ˙’ mdˆa ` n. Sau d¯´o qu´a tr`ınh trˆen d¯u . o . . clˇa . pla . i cho d¯ˆe ´ n khi c`on d¯´ung hai k´y hiˆe . u ho . . p (tˆo ˙’ ng x´ac suˆa ´ ttu . o . ng ´u . ng cu ˙’ ach´ung l`a 1.0). Giai d¯oa . n 2: Xˆay du . . ng t`u . m˜a. M˜a ho´a c´ac k´y hiˆe . u nguˆo ` n: bˇa ´ td¯ˆa ` uv´o . i nguˆo ` n cuˆo ´ i c`ung v`a lˆa ` n ngu . o . . c tro . ˙’ vˆe ` gˆo ´ c. D˜ı nhiˆen c´ac t`u . m˜a nhi . phˆan c´o d¯ˆo . d`ai nho ˙’ nhˆa ´ t (trong nguˆo ` ngˆo ` m hai k´yhiˆe . u) l`a 0 v`a 1. Nhu . H`ınh 6.9 chı ˙’ ra, c´ac t`u . m˜a n`ay d¯u . o . . c g´an cho hai k´y hiˆe . u trˆen cˆo . tbˆen pha ˙’ i-cˆo . t 4 (d˜ı nhiˆen c´o thˆe ˙’ g´an theo th´u . tu . . ngu . o . . cla . i). V`ı 172 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nguˆo ` n ban d¯ˆa ` u Nguˆo ` n thu go . n K´yhiˆe . u X´ac suˆa ´ t 1234 a 2 a 6 a 1 a 4 a 3 a 5 0.4 0.3 0.1 0.1 0.06 0.04 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.4 0.3 0.1 0.1 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.4 0.3 0.2 0.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.4 0.3 0.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.6 0.4 H`ınh 6.8: Qu´a tr`ınh thu go . n nguˆo ` n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nguˆo ` n ban d¯ˆa ` u Nguˆo ` n thu go . n K´yhiˆe . u X´ac suˆa ´ t T`u . m˜a 1234 a 2 a 6 a 1 a 4 a 3 a 5 0.4 0.3 0.1 0.1 0.06 0.04 1 00 011 0100 01010 01011 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.41 0.300 0.1 011 0.1 0100 0.1 0101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.41 0.300 0.2 010 0.1 011 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.41 0.300 0.301 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.60 0.41 H`ınh 6.9: Thu ˙’ tu . c g´an m˜a Huffman. k´yhiˆe . u nguˆo ` n thu go . nv´o . i x´ac suˆa ´ t 0.6 d¯u . o . . cta . obo . ˙’ i hai k´y hiˆe . u nguˆo ` n thu go . n trˆen cˆo . tph´ıa bˆen tr´ai-cˆo . t 3, nˆen hai k´y hiˆe . u n`ay d¯u . o . . c g´an c´ac t`u . m˜a c´o chung tiˆe ` ntˆo ´ 0 v`a sau d¯´o l`a c´ac bit 0 v`a 1. Thao t´ac n`ay d¯u . o . . c thu . . chiˆe . nchod¯ˆe ´ n khi d¯a . td¯ˆe ´ n nguˆo ` n ban d¯ˆa ` u. C´ac t`u . m˜a xuˆa ´ thiˆe . n trong cˆo . tt`u . m˜a cu ˙’ aH`ınh 6.9. D - ˆo . d`ai trung b`ınh cu ˙’ a bˆo . m˜a n`ay l`a L avg = (1)(0.4) + (2)(0 .3) + (3)(0.1) + ((4)0.1) + (5)(0.06) + (5)(0.04) =2.2 bit/k´y hiˆe . u v`a entropy cu ˙’ a nguˆo ` nl`a2.14 bit/k´y hiˆe . u. Suy ra hiˆe . u suˆa ´ tl`a0.973. Phu . o . ng ph´ap Huffman cho bˆo . m˜a tˆo ´ iu . ud¯ˆo ´ iv´o . itˆa . p c´ac k´y hiˆe . u c´o x´ac suˆa ´ tcho tru . ´o . cv´o . i r`ang buˆo . c c´ac k´yhiˆe . ud¯u . o . . c m˜a ho´a liˆen tiˆe ´ p nhau. Sau khi bˆo . m˜a d¯u . o . . c xˆay du . . ng, qu´a tr`ınh m˜a ho´a v`a/hoˇa . c gia ˙’ im˜ad¯u . o . . c thu . . chiˆe . n thˆong qua ba ˙’ ng tra c´u . u. D - ˆay l`a mˆo . tbˆo . m˜a khˆo ´ id¯u . o . . c gia ˙’ i m˜a duy nhˆa ´ tv`at´u . c th`o . i. Ta go . il`abˆo . m˜a khˆo ´ i v`ı 173 mˆo ˜ ik´yhiˆe . utu . o . ng ´u . ng mˆo . t d˜ay c´ac bit. D - ´ol`am˜a t´u . c th`o . i v`ı nh`ın v`ao d˜ay c´ac bit liˆen tiˆe ´ p n`ao d¯´o, ta c´o thˆe ˙’ gia ˙’ i m˜a khi d¯ˆe ´ n cuˆo ´ it`u . m˜a m`a khˆong cˆa ` n pha ˙’ i x´et bit kˆe ´ tiˆe ´ p. D - ´ol`am˜a c´o thˆe ˙’ gia ˙’ i m˜a duy nhˆa ´ t theo ngh˜ıa mˆo . t d˜ay c´ac bit chı ˙’ c´o mˆo . t c´ach gia ˙’ i m˜a. (M˜a c 0 =0,c 1 =1,c 2 =01,c 3 = 10 l`a m˜a khˆong duy nhˆa ´ tv`ı d˜ay c´ac bit 0011 c´o thˆe ˙’ gia ˙’ i m˜a l`a c 0 c 0 c 1 c 1 hoˇa . cl`ac 0 c 2 c 1 ). Do d¯´o mˆo . t d˜ay c´ac bit tu . o . ng ´u . ng thˆong b´ao d¯ u . o . . c m˜a ho´a theo Huffman c´o thˆe ˙’ gia ˙’ i m˜a bˇa ` ng c´ach x´et t`u . ng bit trong d˜ay t`u . tr´ai sang pha ˙’ i. V´o . i m˜a nhi . phˆan trong H`ınh 6.9, duyˆe . t d˜ay 010100111100 cho t`u . m˜a ho . . p lˆe . l`a 01010-tu . o . ng ´u . ng k´y hiˆe . u a 3 . T`u . m˜a ho . . plˆe . tiˆe ´ p theo l`a 011-tu . o . ng ´u . ng a 1 . Tiˆe ´ p tu . c ta d¯u . o . . c thˆong b´ao a 3 a 1 a 2 a 6 . C´ac m˜a c´o d¯ˆo . d`ai thay d¯ˆo ˙’ igˆa ` ntˆo ´ iu . u Xˆay du . . ng m˜a Huffman trong tru . `o . ng ho . . p nguˆo ` ngˆo ` m nhiˆe ` uk´yhiˆe . u l`a tiˆe ´ n tr`ınh rˆa ´ t ph´u . cta . p. X´et nguˆo ` ngˆo ` m J k´yhiˆe . u, ta cˆa ` n J − 2bu . ´o . ctˆo ˙’ ho . . pd¯ˆe ˙’ xˆay du . . ng cˆay nhi . phˆan Huffman v`a J −2bu . ´o . c duyˆe . td¯ˆe ˙’ g´an c´ac t`u . m˜a. Theo kh´ıa ca . nh d¯ˆo . ph´u . cta . p t´ınh to´an th`ı trong mˆo . tsˆo ´ tru . `o . ng ho . . pch´ung ta cˆa ` n hy sinh t´ınh hiˆe . u qua ˙’ trong m˜a ho´a d¯ˆe ˙’ xˆay du . . ng mˆo . t c´ach d¯o . n gia ˙’ nbˆo . m˜a. Ba ˙’ ng 6.3 minh ho . abˆo ´ nbˆo . m˜a c´o d¯ˆo . d`ai thay d¯ˆo ˙’ i nhˇa ` mmu . cd¯´ıch n`ay. Ch´u´y rˇa ` ng d¯ˆo . d`ai trung b`ınh cu ˙’ a m˜a Huffman−h`ang cuˆo ´ icu ˙’ aba ˙’ ng−nho ˙’ ho . nd¯ˆo . d`ai trung b`ınh cu ˙’ a c´ac bˆo . m˜a kh´ac. D - ˆo . d`ai trung b`ınh cu ˙’ am˜atu . . nhiˆen l`a l´o . n nhˆa ´ t. Ho . nn˜u . a entropy cu ˙’ a nguˆo ` nbˇa ` ng 4.0 bit/k´y hiˆe . u trong khi rate cu ˙’ a m˜a Huffman l`a 4.05 bit/k´y hiˆe . u. Mˇa . cd`u khˆong c´o bˆo . m˜a n`ao trong Ba ˙’ ng 6.3 hiˆe . u qua ˙’ bˇa ` ng m˜a Huffman nhu . ng ch´ung d¯u . o . . c xˆay du . . ng dˆe ˜ ho . n. Tu . o . ng tu . . phu . o . ng ph´ap Huffman, ch´ung g´an c´ac t`u . m˜a ngˇa ´ n cho nh˜u . ng k´y hiˆe . uthu . `o . ng xuˆa ´ thiˆe . n. M˜a Huffman chˇa . tcu . t Cˆo . tth´u . nˇam trong Ba ˙’ ng 6.3 minh ho . amˆo . tca ˙’ i biˆen cu ˙’ aphu . o . ng ph´ap Huffman: m˜a Huffman chˇa . tcu . t. M˜a n`ay d¯u . o . . c xˆay du . . ng tu . o . ng tu . . phu . o . ng ph´ap Huffman nhu . ng cho ψ k´yhiˆe . uthu . `o . ng xuˆa ´ thiˆe . n nhˆa ´ t, v´o . i ψ<J.Tˆa ´ tca ˙’ c´ac k´y hiˆe . u c`on la . id¯u . o . . cm˜a ho´a theo c`ung mˆo . t c´ach: c´o chung mˆo . t m˜a tiˆe ` ntˆo ´ v`a sau d¯´o l`a m˜a d¯ˆo . d`ai cˆo ´ d¯ i . nh. Trong Ba ˙’ ng 6.3 sˆo ´ nguyˆen ψ = 12 v`a m˜a tiˆe ` ntˆo ´ xem nhu . t`u . m˜a th´u . 13 trong bˆo . m˜a Huffman. T´u . c l`a c´o mˆo . t“k´yhiˆe . utiˆe ` ntˆo ´ ”v´o . i x´ac suˆa ´ tbˇa ` ng tˆo ˙’ ng x´ac suˆa ´ tcu ˙’ a c´ac k´y hiˆe . u a 13 ,a 14 , ,a 21 ;k´yhiˆe . utiˆe ` ntˆo ´ n`ay c`ung v´o . i12k´yhiˆe . uthu . `o . ng xuˆa ´ thiˆe . n nhˆa ´ t d¯ u . o . . c m˜a ho´a theo Huffman. 9 k´y hiˆe . u c`on la . id¯u . o . . c m˜a ho´a su . ˙’ du . ng t`u . m˜a tiˆe ` ntˆo ´ (trong tru . `o . ng ho . . p n`ay l`a 10) v`a sau d¯´o l`a mˆo . tchuˆo ˜ i 4-bit. Cu . thˆe ˙’ ,k´yhiˆe . u a i ,i≥ 13, 174 c´o t`u . m˜a v´o . i hai b´ıt d¯ˆa ` u10v`abˆo ´ n bit sau tu . o . ng ´u . ng bˆo ´ n bit biˆe ˙’ udiˆe ˜ n nhi . phˆan cu ˙’ a sˆo ´ nguyˆen (i −13). M˜a B M˜a B (cˆo . tth´u . s´au) cu ˙’ aBa ˙’ ng 6.3 l`a m˜a c´o d¯ˆo . d`ai thay d¯ˆo ˙’ igˆa ` ntˆo ´ iu . u. M˜a B gˆa ` ntˆo ´ i u . u khi c´ac x´ac suˆa ´ tk´yhiˆe . u nguˆo ` n c´o da . ng p(a j )=cj −γ ,i=1, 2, ,J, (6.10) trong d¯´o γ l`a hˇa ` ng sˆo ´ du . o . ng n`ao d¯´o v`a c =1/  J j=1 j −γ . Chˇa ˙’ ng ha . n phˆan bˆo ´ d¯ ˆo . d`ai cha . ycu ˙’ amˆo . tbiˆe ˜ udiˆe ˜ n nhi . phˆan trˆen c´ac vˇan ba ˙’ n d¯´anh m´ay c´o da . ng h`am m˜u. Nhu . chı ˙’ ra trong Ba ˙’ ng 6.3, mˆo ˜ it`u . m˜a gˆo ` m hai phˆa ` n: phˆa ` nd¯ˆa ` u l`a c´ac “bit mo . ˙’ rˆo . ng” d¯u . o . . c g´an nh˜an C v`a phˆa ` nth´u . hai gˆo ` m c´ac bit “thˆong tin”. C´ac bit thˆong tin su . ˙’ du . ng m˜a tu . . nhiˆen. C´ac bit mo . ˙’ rˆo . ng d`ung d¯ˆe ˙’ t´ach c´ac t`u . m˜a v`a do d¯´o d¯u . o . . cd¯ˇa . t xen k˜e gi˜u . a0 v`a 1 d¯ˆo ´ iv´o . imˆo ˜ it`u . m˜a trong chuˆo ˜ i. M˜a B trong Ba ˙’ ng 6.3 l`a m˜a B 2 v`ısu . ˙’ du . ng hai bit thˆong tin. Theo c´ach n`ay, thˆong b´ao a 11 a 2 a 7 d¯ u . o . . c m˜a ho´a l`a 001 010 101 000 010 hoˇa . c 101 110 001 100 110 phu . thuˆo . c v`ao bit mo . ˙’ rˆo . ng d¯u . o . . c g´an l`a 0 hoˇa . c 1. Nhˆa . nx´et rˇa ` ng, d¯ˆay khˆong pha ˙’ i l`a m˜a t´u . c th`o . i. M˜a Shift Hai m˜a c´o d¯ˆo . d`ai thay d¯ˆo ˙’ i c`on la . i trong Ba ˙’ ng 6.3 go . il`am˜a shift. M˜a n`ay d¯u . o . . cta . ora theo nguyˆen tˇa ´ c (1) sˇa ´ pxˆe ´ p c´ac k´y hiˆe . u nguˆo ` n sao cho x´ac suˆa ´ t theo th´u . tu . . gia ˙’ mdˆa ` n; (2) chia tˆa . pc´ack´yhiˆe . u nguˆo ` n th`anh c´ac khˆo ´ i c´o k´ıch thu . ´o . cbˇa ` ng nhau; (3) m˜a ho´a c´ac k´y hiˆe . u trong tˆa ´ tca ˙’ c´ac khˆo ´ i giˆo ´ ng nhau; v`a (4) thˆem c´ac k´yhiˆe . ud¯ˇa . cbiˆe . t shift-up v`a/hoˇa . c shift-down d¯ ˆe ˙’ d¯´anh dˆa ´ umˆo ˜ i khˆo ´ i. Khi nhˆa . nd¯u . o . . cmˆo . tk´yhiˆe . u shift-up hoˇa . c shift-down ngu . `o . i gia ˙’ i m˜a s˜e di chuyˆe ˙’ n lˆen hoˇa . c xuˆo ´ ng mˆo . t khˆo ´ itu . o . ng ´u . ng v´o . i khˆo ´ i tham chiˆe ´ ud¯u . o . . cd¯i . nh ngh˜ıa tru . ´o . c. D - ˆe ˙’ xˆay du . . ng m˜a shift 3-bit trong cˆo . tth´u . ba ˙’ ycu ˙’ aBa ˙’ ng 6.3 gˆo ` m21k´yhiˆe . u nguˆo ` n d¯˜a d¯u . o . . csˇa ´ pxˆe ´ p theo th´u . tu . . x´ac suˆa ´ t gia ˙’ m ta chia th`anh ba khˆo ´ i, mˆo ˜ i khˆo ´ igˆo ` mba ˙’ y k´yhiˆe . u. C´ac k´yhiˆe . ucu ˙’ a khˆo ´ itrˆenc`ung d¯u . o . . c xem nhu . khˆo ´ i tham chiˆe ´ u v`a d¯u . o . . cm˜a ho´a th`anh 000 d¯ˆe ´ n 110. T`u . m˜a th´u . t´am (111) khˆong thuˆo . c khˆo ´ i tham chiˆe ´ u m`a d¯u . o . . c d`ung nhu . k´yhiˆe . u shift-up d¯iˆe ` u khiˆe ˙’ ndi . ch xuˆo ´ ng d¯ˆe ˙’ x´ac d¯i . nh c´ac khˆo ´ i c`on la . i(k´y hiˆe . u shift-down khˆong d`ung trong tru . `o . ng ho . . p n`ay). C´ac k´y hiˆe . u trong hai khˆo ´ i c`on la . id¯u . o . . c m˜a ho´a bˇa ` ng c´ach tˆo ˙’ ho . . pmˆo . t hoˇa . chaik´yhiˆe . u shift-up v´o . i c´ac t`u . m˜a trong khˆo ´ i tham chiˆe ´ u. V´ıdu . k´yhiˆe . u nguˆo ` n a 19 d¯ u . o . . c m˜a ho´a l`a 111 111 100. Tu . o . ng tu . . v´o . i m˜a shift Huffman (cˆo . tth´u . t´am cu ˙’ aBa ˙’ ng 6.3). Kh´ac nhau chu ˙’ 175 K´yhiˆe . u X´ac M˜a Huffman B 2 -m˜a M˜a Shift M˜a Shift nguˆo ` n suˆa ´ t nhi . phˆan Huffman chˇa . tcu . t nhi . phˆan Huffman Khˆo ´ i1 a 1 0.2 00000 10 11 C00 000 10 a 2 0.1 00001 110 011 C01 001 11 a 3 0.1 00010 111 0000 C10 010 110 a 4 0.06 00011 0101 0101 C11 011 100 a 5 0.05 00100 00000 00010 C00C00 100 101 a 6 0.05 00101 00001 00011 C00C01 101 1110 a 7 0.05 00110 00010 00100 C00C10 110 1111 Khˆo ´ i2 a 8 0.04 00111 00011 00101 C00C11 111 000 00 10 a 9 0.04 01000 00110 00110 C01C00 111 001 00 11 a 10 0.04 01001 00111 00111 C01C01 111 010 00 110 a 11 0.04 01010 00100 01000 C01C10 111 011 00 100 a 12 0.03 01011 01001 01001 C01C11 111 100 00 101 a 13 0.03 01100 01110 10 0000 C10C00 111 101 00 1110 a 14 0.03 01101 01111 10 0001 C10C01 111 110 00 1111 Khˆo ´ i3 a 15 0.03 01110 01100 10 0010 C10C10 111 111 000 00 00 10 a 16 0.02 01111 010000 10 0011 C10C11 111 111 001 00 00 11 a 17 0.02 10000 010001 10 0100 C11C00 111 111 010 00 00 110 a 18 0.02 10001 001010 10 0101 C11C01 111 111 011 00 00 100 a 19 0.02 10010 001011 10 0110 C11C10 111 111 100 00 00 101 a 20 0.02 10011 011010 10 0111 C11C11 111 111 101 00 00 1110 a 21 0.01 10100 011011 10 1000 C00C00C00 111 111 110 00 00 1111 Entropy 4.0 D - ˆo . d`ai trung b`ınh 5.0 4.05 4.24 4.65 4.59 4.13 Ba ˙’ ng 6.3: C´ac m˜a c´o d¯ˆo . d`ai thay d¯ˆo ˙’ i. 176 . k´yhiˆe . ud¯ˇa . cbiˆe . t shift-up v`a/hoˇa . c shift-down d¯ ˆe ˙’ d¯´anh dˆa ´ umˆo ˜ i khˆo ´ i. Khi nhˆa . nd¯u . o . . cmˆo . tk´yhiˆe . u shift-up hoˇa . c shift-down ngu . `o . i gia ˙’ i. C00C00C00 111 111 110 00 00 1111 Entropy 4.0 D - ˆo . d`ai trung b`ınh 5.0 4.05 4.24 4 .65 4.59 4.13 Ba ˙’ ng 6. 3: C´ac m˜a c´o d¯ˆo . d`ai thay d¯ˆo ˙’ i. 1 76 . nhi . phˆan trong H`ınh 6. 9, duyˆe . t d˜ay 010100111100 cho t`u . m˜a ho . . p lˆe . l`a 01010-tu . o . ng ´u . ng k´y hiˆe . u a 3 . T`u . m˜a ho . . plˆe . tiˆe ´ p theo l`a 011-tu . o . ng ´u . ng