ce.sharif.educe.sharif.edu/courses/92-93/2/ce354-1/resources/...ûÝµþ¤ Úó Û Ü½ ø ü ÂÏ...

ûÝµþ¤Úó ÛÜ½ ø üÂÏ'

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ûÝµþ¤Úó üÂÏ ¤¢ þ (ýÃþ¤õ÷Â) ©ø¤

(Dynamic Programming) þ ýÃþ¤õ÷Â Ûù¤ ûüðÄþø.´¨ ý¥¨ú fÞãõ ÜÿÆõ •.À÷ª Û ú Àþ Ýû ûÜÿÆõÂþ¥ Ýû ú Ûù¤ ýÂ •.¢ªüõ Èõ ýûÜÿÆõ Âþ¥ Û ¤ÂØ °õ ,ÛøÝÆÖ ©ø¤ Û •

ü¨Àì ÀÞ½õ c© 1392 ÀÔ¨ 4 1 Â³õî ü¨Àúõ ýùÀØÈ÷¢


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n ¥ m °îÂ:Ýû¢ ô¹÷ âÞ ÛÞä ú ¤ (

nm

) Û¬ Ýû¡üõ

n

m

=

1, n = m þ m = 0 Âð

(

n−1m

)

+(

n−1

m−1)

, ¤¬ ßþ Âè ¤¢

ü¨Àì ÀÞ½õ c© 1392 ÀÔ¨ 4 2 Â³õî ü¨Àúõ ýùÀØÈ÷¢ ûÝµþ¤Úó ÛÜ½ ø üÂÏ'

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%ÛøÝÆÖ ©ø¤

Combination (n, m). Computes

(

nm

)

using ’+’ only1 if n = m or m = 02 then return 13 else4 return Combination(n−1,m)+Combination (n−1,m−1)



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(11) (11) (11) (11) (11) (11)(10) (10) (10) (10) (10) (10)

(22) (22) (22) (21)(21) (21) (21) (21) (21)(20) (20)(20)

(33) (32) (32) (31) (32) (31) (31) (30)

(42) (42) (41)(43)

(53) (52)

(63)

= 20

1

1

1 1 1 1 1 1 1 1 1 1 1 1

1

1

(63) ý±¨½õ ýÂ ´Èð¥ ´¡¤¢



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ÛÜ½:Àª âÞ ñÞä ¢Àã T (n,m) Âð

T (n,m) =

0, n = m þ m = 0 Âð

T (n− 1,m) + T (n− 1, m− 1) + 1, ¤¬ ßþ Âè ¤¢

T (n, m) =(

nm

)

− 1 î Àþ¢ öüõ(!?Â´¤ Ûù¤)



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ü÷õ ¢Àä ýùÂ¡£ ýÂ ÅþÂõ ¥ ù¢Ôµ¨

C(i, j) =(

ij

) .Ýîüõ ù¢Ôµ¨ C ô÷ (n + 1)× (m + 1) ¢ã ÅþÂõ ×þ ¥

0 1 2 3 4 5 . . m--------------------------

0 11 1 12 1 2 13 1 3 3 14 1 4 6 4 15 1 5 10 10 5 1.....n c


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%þ¤ ×þ ¥ ù¢Ôµ¨ ±¨½õ

Combination (n, m). A is an array of [0..100]

1 A[0]← 12 for i← 1 to n3 do if i ≤ m4 then A[i]← 15 for j ← mini− 1, m downto 16 do A[j]← A[j] + A[j − 1]7 return A[m]



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ñ·õ

0 1 2 3 4 5 6

0 1

1 1 1

2 1 2 1

3 1 3 3 1

4 1 4 6 4 1

5 1 5 9 10 5 1

6 1 6 15 19 15 6 1

7 1 7 21 34 34 21 7

8 1 8 28 35 68 55 28

9 1 9 36 63 103 123 83

10 1 10 45 99 166 226 206



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ÛÜ½ ´¨ ÂÂ Ýµþ¤Úó ßþ ¤¢ âÞ ñÞä ¢Àã

T (n, m) = 1 + 2 + ... + m− 1 + m(n−m) = m(2n−m− 1)/2Âµð¤Ã n ø m ×þ¢Ã÷ Âþ¢Öõ ýÂ üóø (

nm−1) ¥ Â×î m ø n Âþ¢Öõ ü¡Â ýÂ î.´¨ ö ¥



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Â ©ø¤

0 1 2 3 4 . . . m0 11 1 12 1 x 13 1 x x 14 x x x 1. x x x 1. x x x 1. x x x 1. x x x 1. x x x

x xn x


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Â Ýµþ¤Úó

Combination ()1 for i← 1 to n2 do if i ≤ m3 then A[i]← 14 for j ← min(i− 1, m) downto max(1, m− n + i)5 do A[j]← A[j] + A[j − 1]6

(n−m)m =âÞ ñÞä ¢Àã.(n−m)m ≤(

nm

)

− 1 Ýþ¤¢ ø ´¨ Þî ¢Àã ßþ (

nm

)

=(

nn−m

) î ßþ ü¨Àì ÀÞ½õ c© 1392 ÀÔ¨ 4 11 Â³õî ü¨Àúõ ýùÀØÈ÷¢


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©Â

.ÝªøÂÔ ø Ýî ÝÆÖ üãÎì Ýû¡üõ ¤ n ýù¥À÷ ãÎì ×þ •(¼½¬ ¢Àä ûù¥À÷) .´¨ ôÜãõ ãÎì Âû ´Þì •.¢¤À÷ ýþÃû ©Â •?ÝþÂ± ¤ ¢¨ ßþÂÇ Àª ü¤¬ ©Â •



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ñ·õ ãÎì ´Þì

i ýãÎì ñÏ 1 2 3 4 5 6 7 8 9 10

pi ©¥¤ 1 5 8 9 10 17 17 20 24 30.ÀþÂ öüõ ´ó 2n−1 ¤¢ ¤ n ýù¥À÷



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%Àª 4 ýù¥À÷ Âð

59 1 8 5 5

1 1 1 1 1 1 1 1 15 5 18 1


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%üµÈð¥ Ûù¤ø Àª i ñÏ ýãÎì ©øÂê ´Þì (1 ≤ i ≤ n ýÂ) pi Âð,Àª n ñÏ ©Â ¥ ¢¨ ßþÂÇ rn

rn = max1≤i≤npi + rn−i



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ßþ-- ¥ üµÈð¥ Ûù¤

Top-Down-Cut-Rod (p, n)1 if n = 02 then return 03 q ← −∞4 for i← 1 to n5 do q ← max(q, p[i] + Top-Down-Cut-Rod(p, n− i))6 return q



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ÛÜ½ ´Èð¥ ´¡¤¢

13 2 0

2 1 0 1 0 01 0 0 00

4



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ý¤ÂØ ýûÜÿÆõÂþ¥


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(õ¢) ÛÜ½

T (1) = 1

T (n) = 1 +n−1

∑

j=0 T (j) .´¨ 2n ö î



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(Memoized) ßþ-- ¥ Ûù¤

Memoized-Cut-Rod (p, n). r[0..n] is a new array

1 for i← 0 to n2 do r[i]← −∞3 return Memoized-Cut-Rod-Aux(p, n, r))



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Memoized-Cut-Rod-Aux (p, n, r)1 if r[n] ≥ 02 then return r[n]3 if n = 04 then q ← 05 elseq ← −∞6 for i← 1 to n7 do q ←

max(q, p[i] + Memoized-Cut-Rod-Aux(p, n− i, r))8 r[n]← q9 return q



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O(n2) = Â öõ¥


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%(--ßþ ¥) þ Ûù¤

Dynamic-Cut-Rod (p, n). r[0..n] is a new array

1 r[0]← 02 for j ← 1 to n3 do q ← −∞4 for i← 1 to j5 do q ← max(q, p[i] + r[j − i])6 r[j]← q7 return r[n]



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O(n2) = Â öõ¥



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ßµª÷ ýÂ

Dynamic-Cut-Rod-Extended(p, n). r[0..n] and s[0..n] are new arrays

1 r[0]← 02 for j ← 1 to n3 do q ← −∞4 for i← 1 to j5 do if q < p[i] + r[j − i]6 then q ← p[i] + r[j − i]7 s[j]← i8 r[j]← q9 return r and s



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ßµª÷

Print-Cut-Rod-Solution (p, n)1 r, s← Dynamic-Cut-Rod-Extended(p,n)2 while n > 03 do print s[n]4 n← n− s[n]


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%ýÂ ýÜÿÆõ

ùÀª ù¢¢ ` ýù¥À÷ ãÎì ×þ •.ÝþÂ± (´¨¤ ² ¥) xn = ` x1 ¯Ö÷ ýÞû ¥ Àþ ¤ ö •.´¨ ñþ¤ r ö ýÎÖ÷ Âû ¥ 1 ≤ r ≤ n ñÏ ×þ ©Â ýþÃû •¢ª Þî ©Â ýþÃû ÝþÂ± ¤ ùÀª ù¢¢ ãÎì ü±Â •ü¨Àì ÀÞ½õ c© 1392 ÀÔ¨ 4 27 Â³õî ü¨Àúõ ýùÀØÈ÷¢


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? ÜÿÆõÂþ¥xj−1 xi+1 ¯Ö÷ ¥ [xi..xj] ãÎì ýú ©Â Pij.Pij ýÜÿÆõ ýÂ ú ýþÃû Cij



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Cij = mini<k<j

xj − xi + Cik + Ckj



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ûÅþÂõ Â®:Ýî Â® Ýû¤¢ Ýû¡üõ ¤ ÅþÂõ n

M1 ×M2 × · · · ×Mn

di−1 × di : ´¨ ÂÂ Mi ¢ã.¢ª Þî Ûî ýþÃû î Ýû¢ ô¹÷ ü±Â ¤ ûÂ® ßþ Ýû¡üõ


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ñ·õ

M =M1

[10× 20] × M2

[20× 50] × M3

[50× 1] ×M4

[1× 100]ü¨Àì ÀÞ½õ c© 1392 ÀÔ¨ 4 31 Â³õî ü¨Àúõ ýùÀØÈ÷¢


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[10× 20]M1 M2× (M3 ×M4)M1 × (M2× (M3 ×M4))

[20× 100]

[20× 50]

M3 M4

[50× 1]

[10× 100]

[1× 100]

[50× 100]M2 M3 ×M4

C34 = 50× 1× 100 = 5000

C24 = C34 + 20× 50× 100 = 105, 000

C14 = C24 + 10× 20× 100 = 125, 000

M1 × (M2 × (M3 ×M4)) ý±¨½õ ¤¢ ûÅþÂõ Â® ýþÃû ø °Âü¨Àì ÀÞ½õ c© 1392 ÀÔ¨ 4 32 Â³õî ü¨Àúõ ýùÀØÈ÷¢


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[10× 20]C24 = C23 + 10× 20× 1 = 1200 M4M1× (M2 ×M3)

(M1 × (M2×M3))×M4

[10× 1]

[10× 20]

M2 M3

[20× 50] [50× 1]

[20× 1]M1 M2 ×M3

C23 = 20× 50× 1 = 1000

C14 = C13 + 10× 1× 100 = 2200[10× 100]

.(M1 × (M2 ×M3))×M4 ý±¨½õ ¤¢ ûÅþÂõ Â® ýþÃû ø °Âü¨Àì ÀÞ½õ c© 1392 ÀÔ¨ 4 33 Â³õî ü¨Àúõ ýùÀØÈ÷¢


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üµÈð¥ Ûù¤

:ÜÿÆõ Âþ¥

Mij = Mi ×Mi+1 × · · · ×Mj

Cij : Mij ýÂ ú ýþÃû

Cij =

0 if i = jmini≤k<j

Cik + Ck+1,j + di−1dkdj if i < j


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%üµÈð¥ Ûù¤

Recursive-Matrix-Multiplication (d, i, j). Computes C[i, j], the optimal cost of Mi ×Mi+1 × · · · ×Mj

1 if i = j2 then return 03 C[i, j]← ∞4 for k ← i to j − 15 do q ← Recursive-Matrix-Multiplication(d, i, k) +

Recursive-Matrix-Multiplication(d, k + 1, j) +di−1dkdj

6 if q < C[i, j]7 then C[i, j]← q8 return C[i, j]ü¨Àì ÀÞ½õ c© 1392 ÀÔ¨ 4 35 Â³õî ü¨Àúõ ýùÀØÈ÷¢


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ÛÜ½

Recursive-Matrix-Multiplication(d,1, n) ýÂ öõ¥ T (n)

T (1) ≥ 1

T (n) ≥ 1 +n−1

∑

k=1[T (k) + T (n− k) + 1], for n > 1

Å ,¢ªüõ ¤ÂØ ¤ ø¢ T (i) Âû ,Î¤ ßþ ¤¢

T (n) ≥ 2 n−1

∑

i=1 T (i) + nü¨Àì ÀÞ½õ c© 1392 ÀÔ¨ 4 36 Â³õî ü¨Àúõ ýùÀØÈ÷¢


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T (n) = Ω(2n) î Ýîüõ ± ¤ÂØ ø ý¤Áðý ©ø¤ ¥.T (n) ≥ 2n−1 Ýþ¤¢ n ≥ 1 ýÂ î Ýû¢üõ öÈ÷.´¨ ßªø¤ ÂÖµ¨ ýþ,Ýþ¤¢ ÂÖµ¨ ôð ýÂ

T (n) ≥ 2 n−1

∑

i=1 2i−1 + n

= 2 n−2

∑

i=0 2i + n

= 2(2n−1 − 1) + n

= (2n − 2) + n

≥ 2n−1ü¨Àì ÀÞ½õ c© 1392 ÀÔ¨ 4 37 Â³õî ü¨Àúõ ýùÀØÈ÷¢


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ø Àîüõ ü¨¤Â ¤ ÅþÂõ n Â® ýÞû âìø ¤¢ Ýµþ¤Úó ßþ ,ÂÚþ¢ üêÂÏ ¥.¢¤øüõ ´¨¢ ¤ ú ´ó ûö ßß þ Â® ý¤Á ðÃ µ ÷Â þ ø Â® ßØ Þ õ ¢À ã ° ¨ µ õ ö ý þÃ û.Ýû¢üõ ÇþÞ÷ T (n) ¤ö î ´¨ûÅþÂõ,Ýþ¤¢

T (n) =

1 n = 1

n−1∑

i=1 T (i)T (n− i) n > 1

´¨ ÂÂ ø öî ¢Àä ßõ n C(n) î T (n) = C(n− 1) î ¢¢ öÈ÷ öüõ

C(n) =

1n + 1

2nn

= Ω(4n/n

32)ü¨Àì ÀÞ½õ c© 1392 ÀÔ¨ 4 38 Â³õî ü¨Àúõ ýùÀØÈ÷¢ ûÝµþ¤Úó ÛÜ½ ø üÂÏ'

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%ü¨Àì ÀÞ½õ c© 1392 ÀÔ¨ 4 39 Â³õî ü¨Àúõ ýùÀØÈ÷¢


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ý¤ÂØ ýûÜÿÆõÂþ¥

1, 4

1, 1 2, 2

1, 2 3, 4 1, 3 4, 4

3, 3 4, 4

3, 3 4, 4

1, 1 2, 4

2, 2 3, 4 2, 3 4, 4

2, 2 3, 3 2, 2

1, 1 2, 3 1, 2 3, 4

3, 3 1, 1 2, 2

Recursive-Matrix-Multiplication(d,1,4) ýÂ ûü÷¡Âê ´¡¤¢ü¨Àì ÀÞ½õ c© 1392 ÀÔ¨ 4 40 Â³õî ü¨Àúõ ýùÀØÈ÷¢


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þ Ûù¤?Â ,¢¤¢ þ Ûù¤ ÜÿÆõ:ÜÿÆõÂþ¥

Mij = Mi ×Mi+1 × · · · ×Mj!À÷ª Û ú Àþ Ýû ûÜÿÆõÂþ¥ ýÞû ÜÿÆõ ýú Û ýÂ î ´¨ üúþÀø Cii = 0 :Ýþ¤¢ ,Àª Mij ýú ýþÃû Cij Âð

Cij = mini≤k<j

Cik + Ck+1,j + di−1dkdj



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þ Ýµþ¤Úó

Dybnamic-Matrix-Multiplicationi(d)1 for i← 1 to n2 do C[i, i]← 03 for l← 2 to n4 do

. l = number of multiplied matrices5 for i← 1 to n− l + 16 do j ← i + l − 1

. solving Mij

7 for k ← i to j − 18 do C[i, j] ← minC[i, k] + C[k + 1, j] + d[i − 1] ∗

d[k] ∗ d[j]9 R[i, j]← the value of k that makes the minimumü¨Àì ÀÞ½õ c© 1392 ÀÔ¨ 4 42 Â³õî ü¨Àúõ ýùÀØÈ÷¢ ûÝµþ¤Úó ÛÜ½ ø üÂÏ'

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%ú ý¤ÁðÃµ÷Â ßµª÷

PrintResults (i, j)1 if i 6= j2 then k ← R[i, j]3 Print ′(′

4 PrintResults(i, k)5 Print ′×′

6 PrintResults (k + 1, j)7 Print ′)′

O(n2) ýÑê ø O(n3) öõ¥ :ÛÜ½



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1 ñ·õ

M =M1

[10× 20] × M2

[20× 50] × M3

[50× 1] ×M4

[1× 100]

1 2 3 41 0 10000, 1 1200, 1 2200, 32 0 1000, 2 3000, 33 0 5000, 34 0



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ñ·õ :Âþ¥ ýûù¥À÷ ÅþÂõ Çª Â®

M1

[30× 35]

×M2

[35× 15]

×M3

[15× 5]

×M4

[5× 10]

×M5

[10× 20]

×M6

[20× 25]



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C[i,j]

1

2

3

4

5

6

6

5

4

3

2

1

M1 M2 M3 M4 M5 M6

j i

15, 125

11, 875 10, 500

9, 375 7, 125 5, 375

7, 875 4, 375 2, 500 3, 500

15, 750 2, 625 750 1, 000 5, 000

0 0 0 0 0 0

2

3

4

5

6

R[i,j]

5

4

3

2

1

1

j i

2 3 4 5

1 3 3 5

3 3 3

33

3

((M1 × (M2 ×M3))× ((M4 ×M5)×M6))

C[2, 5] = min

C[2, 2] + C[3, 5] + d1d2d5 = 0 + 2500 + 35 ∗ 15 ∗ 20 = 13000,C[2, 3] + C[4, 5] + d1d3d5 = 2625 + 1000 + 35 ∗ 5 ∗ 20 = 7125,C[2, 4] + C[5, 5] + d1d4d5 = 4375 + 0 + 35 ∗ 10 ∗ 20 = 11375

= 7125.


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(Memoization) ý¤³¨ ÂÏ¡ ©ø¤

÷¤î¤î ø ßþ ¥ : üµÈð¥ •ûÜÿÆõÂþ¥ Û¬ ýùÂ¡£ ø ßþ ¥ : þ •ý¤ÂØ ¤î ö¢À÷ ô¹÷ üóø ßþ ¥ : ý¤³¨ÂÏ¡ •ü¨Àì ÀÞ½õ c© 1392 ÀÔ¨ 4 47 Â³õî ü¨Àúõ ýùÀØÈ÷¢


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Memoized-Matrix-Multiplication (d)1 n← length[d]− 12 for i← 1 to n3 do for j ← i to n4 do C[i, j]← ∞



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Lookup-Matrix (p, 1, n)1 if C[i, j] <∞2 then return C[i, j]3 if i = j4 then C[i, j]← 05 elsefor k ← i to j − 16 do q ← Lookup-Matrix(d, i, k) +

Lookup-Matrix(d, k + 1, j) + di−1dkdj

7 if q < C[i, j]8 then C[i, j]← q9 return C[i, j]

ÅþÂõ ýþ¤¢ (Θ(n2) î Â ,´¨ Θ(n3) ý±Âõ ¥ þ ©ø¤ À÷õ Ã÷ ©ø¤ ßþ.¢ªüõ Æ Θ(n) ý±Âõ ¤ Âû ø ¤×þ ÍÖê ×þ Âû C



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À½õ üãÜ®À ýú ýÀ¶Ü·õ

v1v2

v3v0

v4v5v6 v1

v2v3

v0v4

v5v6

ö ýÀ¶Ü·õ á÷ ø¢ ø À½õ üãÜ®À ×þ


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%ÜÿÆõ ýÀñõÂê

´¨ ùÀªù¢¢ P0,n−1 =< v0, v1, . . . , vn−1 > À½õ üãÜ®- n •ôi §b¤ Êµ¿õ vi •ÀµÆû P á® < vi, vi−1 > ø < vi, vi+1 > •À÷Âðüõ ¤Âì P Û¡¢ ¤¢ ûÂÎì .´¨ ÂÎì ,Àª±÷ âÜ® î < vi, vj > •?Â .´ ¨ n − 3 ûÂ Î ì ¢À ã ø n − 2 ýÀ ¶ Ü · õ Â û ý û¶ Ü · õ ¢À ã •(3(n− 2) = n + 2(n− 3))¢Âðüõ ¤Âì ¶Ü·õ ×þ ¤¢ âÜ® Âû •ü¨Àì ÀÞ½õ c© 1392 ÀÔ¨ 4 51 Â³õî ü¨Àúõ ýùÀØÈ÷¢


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À½õ üãÜ®À ×þ v0, v1, . . . , vn−1 §¤ Êµ¿õ :ÜÿÆõ ý¢ø¤ø.Àª Þî öÈóÏ ø Àî ýÀ¶Ü·õ ¤ üãÜ®À î üþûÂÎì :üøÂ¡.¢Âî ÓþÂã Û¬ ýû¶Ü·õ þ ûÂÎì ,ûâÜ® ýø¤ Â w ö¥ø â öüõ¢ª Þî ûö¥ø âÞ Û¬ ö ¤¢ î ýýÀ¶Ü·õ :ú ýÀ¶Ü·õÀª ö¥ø â ×þ À÷üõ ûÂÎì ñÏ:¢ªüõ ÓþÂã Âþ¥ ¤¬ ¶Ü·õ Âû ýø¤ Â û¶Ü·õ ö¥ø

w(∆vivjvk) = vivj + vjvk + vivk

.´¨ vj ø vi ß ü¨ÀÜì ýÜ¬ê vivjü¨Àì ÀÞ½õ c© 1392 ÀÔ¨ 4 52 Â³õî ü¨Àúõ ýùÀØÈ÷¢


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A1

A2

A3

A4 A5

v1

v2

v3

v0

v4

v5v6

A1

A2 A3 A4

A5 A6

A6

.À½õ üãÜ®À ×þ ýÀ¶Ü·õ ø ûÅþÂõ Â® ýûÜÿÆõ ÂÒ

((A1(A2A3))(A4(A5A6)) : ÂÒµõ



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ú ýÀ¶Ü·õ ûÅþÂõ Â® ýÜÿÆõ ÛþÀ±

d0, d1, . . . , dn ýûù¥À÷ A1 ×A2 × ...×An :ý¢ø¤øú ýÀ¶Ü·õ ¤ö ø Ýþ¥¨üõ v0, v1, . . . , vn §¤ À½õ üãÜ® n + 1 ×þ •Ýîüõ

< vi−1vi > âÜ® ÂÒµõ Ai ÅþÂõ •A1 × A2 × ...×An ¢ªüõ ±¨½õ w(∆vivjvk

) = didjdk ¶Ü·õ ö¥ø •Ai+1 ×Ai+1 × ...×Aj ýûÅþÂõ Â® ´¨ ÂÒµõ (i < j) < vi, vj > ÂÎì •ýÂ §¤ ´¡¤¢ ë ê â ß µ êÂ ð Â Ñ ÷ ¤¢ P üã Ü®À ú ýÀ ¶ Ü · õ •.Àîüõ ¢¹þ ¤ A1A2...An ýú ý¤ÁðÃµ÷Âü¨Àì ÀÞ½õ c© 1392 ÀÔ¨ 4 54 Â³õî ü¨Àúõ ýùÀØÈ÷¢ ûÝµþ¤Úó ÛÜ½ ø üÂÏ'

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ú ýÀ¶Ü·õ ýÜÿÆõ Û ýÂ ûÅþÂõ Â® Àî ¥ ù¢Ôµ¨

v0, v1, . . . , vn−1 §¤ P üãÜ® n :ý¢ø¤ø.ÝþÂðüõ ÂÑ÷ ¤¢ ¤ A1 ×A2 × ...×An−1 ýûÅþÂõ Â® •ÂÑ ÷ ¤¢ w(Ai) = vi−1vi ¤ö ö¥ø ø < vi−1vi > â Ü® ´¨ ÂÒ µ õ Ai ÅþÂ õ •.ÝþÂðüõø .Ýîüõ ±¨½õ w(Ai) + w(Ai+1) ÂÂ Ai × Ai+1 ÅþÂõ ø¢ Â® ýþÃû •.ÝþÂðüõ (ÂÎì ñÏ) w(Ai × Ai+1) = vi−1vi+1

< vi−1vi+1 > âÜ® ´¨ ÂÒµõ Ai ×Ai+1 ÅþÂõ •Ai+1 ×Ai+1 × ...×Aj ýûÅþÂõ Â® ´¨ ÂÒµõ (i < j) < vi, vj > ÂÎì •ü¨Àì ÀÞ½õ c© 1392 ÀÔ¨ 4 55 Â³õî ü¨Àúõ ýùÀØÈ÷¢


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.Ýîüõ Â Âç ßþ ¤ ûÅþÂõ Â® Àî •



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ÝÖµÆõ Ûù¤

i ≤ j + 2 ýÂ Pi,j =< vi, vi+1, . . . , vj > :ÜÿÆõÂþ¥ •(P1,n ýÂ ÂÚõ) ´¨ ÂÎì < vi, vj > •

Pij ýú ýþÃû Cij •(i < k < j) .< vi, vj > âÜ® ü·Ü·õ ÛÖõ §b¤ vk •



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:Ýþ¤¢

Ci,j =

0 i ≤ j ≤ i + 2

mini<k<jCik + Ckj + dik + dkj, j > i + 2

.dij = 0 ,Àª âÜ® < vi, vj > Âð .´¨ < vi, vj > ÂÎì ýù¥À÷ ÂÂ dij.´¨ üúþÀ O(n3) þ Ýµþ¤Úó


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%íÂµÈõ ýó±÷¢Âþ¥ ßþÂï¤Ã

b

a b c b d a b

d bcfÀî ýó± ÷¢ Âð ´¨ X =< x1, x2, · · · , xm > ýó± ÷¢Âþ¥ Z =< z1, z2, · · · , zk >ýÂ îý¤Ï Àª µª¢ ¢ø X Â¬ä ýûÅþÀ÷ ¥ < i1, i2, · · · , ik > ý¢ã¬.zi = xij :Ýª µª¢ j = 1, . . . , ký ó ± ÷¢ î ´¨ X =< a, b, c, b, d, a, b > ¥ ó ± ÷¢Â þ¥ × þ Z =< b, c, d, b > f · õ.Àªüõ < 2, 3, 5,7 > ¯Âõ ýûÅþÀ÷ü¨Àì ÀÞ½õ c© 1392 ÀÔ¨ 4 59 Â³õî ü¨Àúõ ýùÀØÈ÷¢


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Longest Common Subsequence) "íÂµÈõ ýó±÷¢Âþ¥ ßþÂï¤Ã' Z ö¢Âî À :ÜÿÆõ.´¨ Y ø X ýùÀª ù¢¢ ýó±÷¢ ø¢ ýÂ ((LCS)

c

a b c b d a b

b d ba a

c

a b c b d a b

b d ba a



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Y =< y1, y2, · · · , yn > ø X =< x1, x2, · · · , xm > ýó±÷¢ ø¢ :ý¢ø¤ø.ó±÷¢ ø¢ ßþ LCS :üøÂ¡

diff file1 file2 :¢Â¤î



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ú ¤µ¡¨Âþ¥

xm. . .

y1 y2 yn−1 yn

. . .

x1 x2 xm−1xm

. . .

y1 y2 yn−1 yn

. . .

x1 x2 xm−1


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xm. . .

y1 y2 yn−1 yn

. . .

x1 x2 xm−1

xm. . .

y1 y2 yn−1 yn

. . .

x1 x2 xm−1



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ó±÷¢ ×þ :X =< x1, x2, · · · , xm >

i = 1..m ýÂ X (prefix) À÷È ßõ i :Xi ≡< x1, x2, · · · , xm >.X4 =< A,B,C, B > :Ýþ¤¢ ,X =< A, B,C,B,D,A,B > ýÂ f ·õ



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ý¢ø¤ø ýû ó ± ÷¢ Y =< y1, y2, · · · , yn > ø X =< x1, x2, · · · , xm > Âð 1 ýÌì:Ýþ¤¢ Àª Z =< z1, z2, · · · , zk > ø¢ ßþ LCS ø Àª.´¨ Yn−1 ø Xm−1 LCS ÂÂ Zk−1 ø zk = xm = yn Ýþ¤¢ ,xm = yn Âð .1.´¨ Y ø Xm−1 LCS ÂÂ Z î ÝþÂðüõ ¹µ÷ zk 6= xm ¥ ùðö ,xm 6= yn Âð .2.´¨ X ø Yn−1 LCS ÂÂ Z î ÝþÂðüõ ¹µ÷ zk 6= yn ¥ ùðö ,xm 6= yn Âð .3



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üµÈð¥ Ûù¤.¢ªüõ ÓÜµ¿õ ýûÜÿÆõÂþ¥ ý¤ÂØ Û Â¹õ üµÈð¥ Ûù¤


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þ Ûù¤,Àª Yj ø Xi ýÂ LCS ñÏ c[i, j] Âð

c[i, j] =

0 if i = 0 or j = 0,c[i− 1, j − 1] + 1 if i, j > 0 and xi = yj,max(c[i, j − 1], c[i− 1, j]) if i, j > 0 and xi 6= yj.



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for i := 1 to m do c[i, 0] := 0;

for j := 1 to m do c[0, j] := 0;

for i := 1 to m do

for j := 1 to n do

if xi = yj then begin

c[i, j] := c[i− 1, j − 1] + 1;

b[i, j] := ‘ ’

end

else if c[i− 1, j] ≥ c[i, j − 1]

then begin

c[i, j] := c[i− 1, j];

b[i, j] := ‘ ↑ ’

end

else begin

c[i, j] := c[i, j − 1];

b[i, j] := ‘← ’

end

O(mn) Ã÷ ö üêÂÊõ ýÑê öÃõ ø O(mn) ü÷õ¥ ý±Âõ ¥ ùÀª ¤ Ýµþ¤Úóü¨Àì ÀÞ½õ c© 1392 ÀÔ¨ 4 68 Â³õî ü¨Àúõ ýùÀØÈ÷¢


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.´¨



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Y =< B,D,C,A, B,A > ø X =< A, B,C,B,D,A, B > :ñ·õ

0

0

0

0

0

0

0

0

0 0 0 0 0 0

0 0 0

1

1

1

1

1

1

1

1

1

1 1

2

2

2

2 2

1 1 1

2222

2 2

22

2

2 3

3 3

3 3

33

4 4

4

↑ ↑ ↑

↑

↑↑

↑↑↑

↑↑↑↑

↑↑↑↑

↑↑↑↑

←

←←←

←

←

ab

a

a

bb

b

b

b

c

c

d

d

xi

yj

0

0

1

1

2

2

3

3

4

4

5

5 6

6

7

ij

↑↑

↑


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LCS ßµ¡¨ü ÷ ¡Â ê ¤ LCS ö ü õ b Å þÂ õ ¤¢ ùÀ ªùÂ ¡£ ä Ï .´¡¨ Print-LCS(b, X,m, n)



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Print-LCS(b, X, i, j)

if (i = 0 or j = 0)

then return;

if b[i, j] = ‘ ’

then begin

Print-LCS (b, X, i− 1, j − 1);

print xi

end

else if b[i, j] = ‘ ↑ ’

then Print-LCS (b, X, i− 1, j)

else Print-LCS (b, X, i− 1, j).´¨ O(m + n) ý±Âõ ¥ Ýµþ¤Úó ßþ



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(Optimal BST) ú ýø´Æ üþø¢ø¢ ´¡¤¢

Â¬ ä ßþ ýÂ ø´Æ ñÞ µ ø a1 < a2 < · · · < a7 ÂÊä 7 Ûõª ´¡¤¢Àª 624 324 , 524 , 424 , 324 , 124 , 224 ÂÂ °Â

a1 a2 a4 a5

a3

a7

a6ßÚ÷õ ö¥µõ÷ ´¡¤¢5826 ýø´Æ öõ¥

a1 a2 a4

a3 a6

a5 a7öõ¥ ßÚ÷õ ö¥µõ ´¡¤¢6426 ýø´Æ



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%û¢Þ÷ ñøÀ ,÷¡µî :ûñ·õ

b6

b5b3b2 a3b0

else

fora5

a7b1a2

do

begina1

b4a4

end

then

a6

b8

if

b7

a8

while

û¢Þ÷ ñøÀ ýÂ ø´Æ üþø¢ø¢ ´¡¤¢ ×þ


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%üÜî ´ó ¤¢ ÜÿÆõ

ü¤¡ ø üÜ¡¢ ÂÊä :(bi ø ai) Õêõ÷ ø Õêõ ýø´Æ

b0 < a1 < b2 < · · · < ai−1 < bi < ai < · · · < bn−1 < an < bn

n + 1 ÂÂ ÂÊä n ø´Æ üþø¢ø¢ ´¡¤¢ ×þ ýÂ ,ü¤¡ Â¬ä ¢Àã .Ýó.´¨



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ÜÿÆõ ý¢ø¤ø Õì¢ ÓþÂã´¡¤¢ Â¬ä a1 < a2 < · · · < an /(i = 1..n) ai ýÂ Õêõ ø´Æ ñÞµ pi /(i = 1..n− 1) ai < bi < ai+1 Âð ,bi ýÂ Õêõ÷ ýø´Æ ñÞµ qi /

b0 < a1 ýÂ Õêõ÷ ýø´Æ ñÞµ q0 /

an < bn ýÂ Õêõ÷ ýø´Æ ñÞµ qn /



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ÜÿÆõ üøÂ¡

öõ¥ Í¨µõ î Àþ¥Æ ú ýø´Æ üþø¢ø¢ ´¡¤¢ ×þ û qi ø û pi ßµª¢ ¢ª Þî ö ¤¢ ( Õêõ ÷ þ Õêõ¥ Ýä) ø´Æ¤ÀÖõ ßþ

n∑

i=1 pi(1 + depth(ai)) +n∑

i=0 qi(depth(bi))

n∑

i=1 pi +n∑

i=0 qi = 1 î



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ÜÿÆõÂþ¥ ÓþÂã

Tij ýÜÿÆõ Âþ¥ •qi, pi+1, qi+1, . . . , pj, qj ý¢ø¤ø ai+1 < · · · < aj ýÂ OBST ´¡¤¢.´¨ üÜ¬ ýÜÿÆõ T0n

Tij ýþÃû : Cij •

Cij =j

∑

r=i+1 pr[depht(ar) + 1] +j

∑

r=0 qr[depth(br)]


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.Àª Tij ýÈþ¤ ak Âð Cij : C(k)ij •:Tij ¤¢ qi ø pi ýûñÞµ áÞ¹õ ÂÂ ,Tij ´¡¤¢ ö¥ø : wij •

wij = qi +j

∑

r=i+1(pr + qr)

Tij ýÈþ¤ rij •ü¨Àì ÀÞ½õ c© 1392 ÀÔ¨ 4 79 Â³õî ü¨Àúõ ýùÀØÈ÷¢


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ai+1 < · · · < aj ýÂ Tij ´¡¤¢ Âþ¥ak

Tij

Tk,jTi,k−1

i < k ≤ j



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Tij ýÜÿÆõ Âþ¥ Û:Ýþ¤¢ ,Àª Èþ¤ (i < k ≤ j ýÂ) ak Âð

C(k)ij = (Ci,k−1 + wi,k−1) + (Ckj + wkj) + pk

= Ci,k−1 + Ckj + wij :Ýþ¤¢ ø

Cij = mini<k≤j

C(k)ij

cii = 0 ø wii = qi ⇐= Ýþ¤¢ Tii ýÂ áøÂª ¤¢



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Ýµþ¤Úó

OBST (p1, . . . , pn, q0, . . . , qn)1 for i← 0 to n2 do wii ← qi

3 cii ← 04 for l← 1 to n5 do for i← 0 to n− l6 do j ← i + l7 wij ← wi,j−1 + pj + qj

8 cij ← mini<k≤j

ci,k−1 + ckj + wij

9 rij ← the k for which above is mininimum


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ÛÜ½.´¨ Θ(n3) ¥ ëê Ýµþ¤Úó



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¢Âî Û O(n2) ¤¢ ¤ Ýµþ¤Úó ßþ öüõ

,Àª Tij ýÜÿÆõ Âþ¥ ýú ýÈþ¤ ýù¤Þª rij Âðri,j−1 ≤ rij ≤ ri+1,j

.Àîüõ O(n2) ¤ Ûù¤ þ Ýµþ¤Úó ¤¢ ëê ýÎ¤ ¥ ù¢Ôµ¨



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ñ·õ

n = 4

a1 < a2 < a3 < a4

p1 = 14

p2 = 18

p3 = p4 = 116

q0 = 18

q1 = 316

q2 = q3 = q4 = 116



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ûñøÀ:Àþüõ Âþ¥ ñøÀ ¤¢ ùÀªµÔð Ýµþ¤Úó ÓÜµ¿õ ÛÂõ


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0 1 2 3 4

c00 = 0 c01 = 9 c02 = 18 c03 = 25 c04 = 330 w00 = 2 w01 = 9 w02 = 12 w03 = 14 w04 = 16

r01 = a1 r02 = a1 r03 = a1 r04 = a2

c11 = 0 c12 = 6 c13 = 11 c14 = 181 w11 = 3 w12 = 6 w13 = 8 w14 = 10

r12 = a2 r13 = a2 r14 = a2

c22 = 0 c23 = 3 c24 = 82 w22 = 1 w23 = 3 w24 = 5

r23 = a3 r24 = a4

c33 = 0 c34 = 33 w33 = 1 w34 = 3

r14 = a4

c44 = 04 w44 = 1



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ñ·õ ´¡¤¢

T01

a1

T04

a2

T24

a4

T23

a3



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(Knapsack) üµÈóî ýûÜÿÆõ:ý¢ø¤ø¤ ö¥ø Àø M ÛÞ üþ÷ üµÈóî ×þ •¤ á÷ N •

Ni ÂÂ i á÷ ¤ ¢Àã •Wi ô i á÷ ¤ ö¥ø •

Ci ô i á÷ ¤ ©¥¤ •.ÀµÆû ¼½¬ ¢Àä Wi ø M :Âê



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û¤ Âð îý¤Ï Ýî Â öØõ À þ f õî û¤ ßþ ¤ üµÈóî Ýû¡ üõ.¢ª È û¤ ©¥¤ áÞ¹õ ,Àª µª¢ ©¥¤


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¢ª Â üµÈóî ø ,Ni = 1 ,Ci = 0 :ñø ´ó.´¨ NP-Complete ýÜÿÆõ ×þ î ´¨ ¤úÈõ 0-1-Knapsack ÜÿÆõ ßþ



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(Backtracking) ¢ÂðÅ Ûù¤

N i ýû¤ ¥ M ýù¥À÷ üµÈóî ö¢Âî ÂKnapsack (M, i)

. returns true if M can be filled from loads i to N1 if M = 02 then return true3 if M < 0 or i > N4 then return false

. load i is a candidate5 if Knapsack (M −Wi, i + 1)6 then Print i, Wi7 return true8 elsereturn Knapsack (M, i + 1)



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.Àª µªÀ÷ ÜÿÆõ î ´¨ üõÚû ´ó ßþÂÀ

T (n) = 2T (n− 1) + Θ(1)⇒ T (n) = Θ(2n)



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÷¤î¤î Û ù¤

BlindKnapSack ()1 for i← 0 to 2N − 12 do find the N-bit binary representation of i3 find the loads with 1’s in binary form of i4 if the sum of weights of the selected loads is equal to M5 then this is one solution

.Àîüõ ÛÞä Ýû À÷õ Ýµþ¤Úó ø¢ ßþ ´ó ßþÂÀ ¤¢


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þ Ûù¤

i 1 ýû¤ ¥ j üµÈóî ö¢Âî Â ,S[i, j] :ÜÿÆõÂþ¥?À÷ª Û ú Àþ ûÜÿÆõÂþ¥ ýÞû þÝû ÜÿÆõÂþ¥ ö ,Ýþ¤¢Â ¥ ¤ ×þ ,Àª ùÀª Û ú ÜÿÆõ Âð .Ü :.¢ª Û ú Àþ

Wi ≤M :Âê:SM×N ÅþÂõ´¨ k > 0 ÂÂ ÷Âð ø ´¨ ÂÔ¬ ¤ÀÖõ ßþ ,¢Âî Â öµ÷ ¤ j üµÈóî Âð :S[i, j].Ýû¢üõ ¤Âì üµÈóî ¤¢ î ´¨ ý¤ ßþÂ¡ ýù¤Þª k î



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0 1 j k = j + Wi M

i

N

1

S[i, k] = S[i− 1, k] ∨ S[i− 1, j]

i− 1

?¢Âî Â i 1 ¥ ýû¤ ¤ j ýù¥À÷ üµÈóî öüõ þ :S[i, j]



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0 1 M

N

1

W1ñø ÂÎ¨ ,áøÂª



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%

N

1

W1W2 W1 + W20 1 M

ôø¢ ÂÎ¨


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%

N

1

W1W2 W1 + W20 1 M

ô¨ ÂÎ¨



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N

1

0 1 M

Â¡ ÂÎ¨



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0 1 j k = j + Wi M

i

N

1

:ö¢Âî Â ÂÚþ¢ á÷

S[i, k]← i ,Àª µªÀ÷ ¤ÀÖõ S[i, k] Âð



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ÅþÂõ ö¢Âî Â(k = j + Wi Âê) :Ýµþ¤Úó Âª

S[1,W1] = 1 :Ýþ¤¢ ñø ÂÎ¨ ¤¢ •.¢ªüõ Â i− 1 ÂÎ¨ §¨ Â i ÂÎ¨ •.¢ªüõ ü³î i ÂÎ¨ ¤¢ i− 1 ÂÎ¨ Âþ¢Öõ ýÞû Àµ •.Àª ùÀÈ÷ Â üÜ±ì ýû¤ Í¨ S[i, k] Âð ,Ýîüõ Â ¤ S[i, k] ,S[i, j] ¥ •.Ýîüõ Â i ¤ ¢Àä ×þ ÍÖê ¤ S[i, k] ,S[i, j] <> i þ ,j = 0 Âð •


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KnapSack-1 (M, Weight)1 N ← length[Weight]2 for i← 1 to N3 do for j ← 0 to M −Wi

4 do S[i, j]← S[i− 1, j]5 k ← j + Wi

6 if (j = 0) or((S[i, j] > 0) and (S[i, j] <> i) and (S[i − 1, k] =

0))7 then . we can fill S[i, k], putting load i last8 S[i, k]← i



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.¢¢ ô¹÷ öüõ ¤ ¤î ßÞû þ¤ ×þ KnapSack-1 (M, Weight)

1 N ← length[Weight]2 for i← 1 to N3 do for j ← 0 to M −Wi

4 do k ← j + Wi

5 if (j = 0) or((S[j] > 0) and (S[j] <> i) and (S[k] = 0))

6 then . we can fill S[i, k], putting load i last7 S[k]← i



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Print-One-Result (S, j). print the one solution for knapsack of size j

1 if j = 02 then return3 k ← S[j]4 if k > 05 then Print k6 Print-One-Result (j −Wk)

i ¤ ¥ ù¢Ôµ¨ üØþ) Àª µª¢ ×þ ¥ Ç S[i, j] ýÜÿÆõ Âð î ßþ Ûó¢ëê Ûù¤ ,Ýû¢üõ S[i− 1, j] ¤ ´þóø õ (ö ¥ ÂÝî ýû¤ ýÂÚþ¢ ø.j = k þ S[j − k] < k fÞµ ,S[j] = k Âð üãþ .´¨ ´¨¤¢



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Â f õî ø ,1 < Ni <∞ ,Ci = 0 :(üÜî) ôø¢ ´óø S[i − 1, j] ý û Ü ÿ Æ õÂ þ¥ ,S[i, j] ý ú Û ýÂ .S[i, j] : Ü ÿ Æ õÂ þ¥ (Ý ú õ).Àª ùÀª Û ú Àþ Ýû S[i− 1, j −Wi].¢Âî Û ¢ªüõ Ýû þ¤ ×þ ¤ö ø ¢¤¢ þ Ûù¤ ÜÿÆõ Å:´¨ Ôóõ ø¢ Ûõª S[j] ýþ¤¢ ´ó ßþ ¤¢üµÈóî ßþ Âð .¢Âðüõ ¤Âì j üµÈóî ¤¢ î ý¤ ßþÂ¡ ýù¤Þª :last(S[j]) •.´¨ ÂÔ¬ ÂÂ Ôóõ ßþ ,¢Âî Â öµ÷ ¤.üµÈóî ßþ ¤¢ last(S[j]) ¤ ¥ ùÀª ù¢Ôµ¨ ¢Àã :Num(S[j]) •.i á÷ ¥ ¤ ¢Àã :Ni


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Ýµþ¤Úó Âª¤¬ ßþ ¤¢ .Ýîüõ Â ¤ ÅþÂõ ô i ÂÎ¨ i Âû ýÂ¤Âì ¤ S[k] üÎþÂª ´½ S[j] ¥ ,k = j + Wi Âê ø ,j = 1..M −Wi ýÂ •.Ýîüõ Â ô i ¤ ¥ ÂÚþ¢ ¢Àä ×þ ö¢¢.¢ªüõ Â ,Àª ùÀÈ÷ Â f ±ì î ü¤¬ ¤¢ ÍÖê S[k] •¤ ßþÂ¡ üóø Àª Â S[j] þ ,j = 0 Âð ,¢ªüõ ù¢Ô µ¨ ¤ ß óø ýÂ i ¤ •.Àª±÷ i ö ¤¢ ùÀª ù¢Ôµ¨.Àª ¢õ üêî ¢Àã Âð ,¢ªüõ ù¢Ôµ¨ f¢À¹õ i ¤ •.À÷Âðüõ ¤Âì üµÈóî ¤¢ öªù¤Þª °Â û¤ :



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KnapSack-2 (M, Weight)1 N ← length[Weight]2 for j ← 1 to M3 do last(S[j])← 04 for i← 1 to N5 do for j ← 1 to M −Wi

6 do k ← j + Wi

7 if last(S[k]) = 08 then . trying to fill S[k]9 if (j = 0) or ((last(S[j]) > 0)

10 then if last(S[j]) 6= i))11 then . use load i for the first time12 last(S[k])← i13 Num(S[k])← 114 elseif Num(S[j]) < Ni

15 then last(S[k])← i16 Num(S[k])← Num(S[j]) + 1ü¨Àì ÀÞ½õ c© 1392 ÀÔ¨ 4 108 Â³õî ü¨Àúõ ýùÀØÈ÷¢


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Print-One-Result (S, j). print the one solution for knapsack of size j

1 if j = 02 then return3 k ← last(S[j])4 if k > 05 then Print k6 Print-One-Result (j −Wk)



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Â f õî ø ,Ni =∞ ,Ci = 0 :ô¨ ´ó.¢ª ù¢Ôµ¨ ¤ ×þ ¥ Ç À÷üõ i á÷ ¤ ´ó ßþ ¤¢

KnapSack-3 (M, Weight)1 N ← length[Weight]2 S ← 03 for i← 1 to N4 do for j ← 0 to M −Wi

5 do k ← j + Wi

6 if j = 0 or S[j] 6= 07 then S[k]← i

.Àîüõ ¤î ´¨¤¢ Ã÷ ´ó ßþ ¤¢ Print-One-Result ýþø¤


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üµÈóî ýÜÿÆõ ×þ (Àª Þî ûØ¨ ¢Àã Ýû¿÷ Âð) ñ ö¢Âî ¢Â¡ ýÜÿÆõ.´¨ ni =∞



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Â f õî ø Ci > 0 ,Ni = 1 :ô¤ú ´óø S[i− 1, j] ýûÜÿÆõÂþ¥ ýû ÝÞþÃîõ ÂÂ S[i, j] ýÜÿÆõ ýú S[i− 1, j −Wi].À÷ª Û ú Àþ ø ¢ªüÞ÷ ù¢Ôµ¨ i ¤ ¥ ÜÿÆõÂþ¥ ø¢ ßþ ¤¢þ Ûù¤ :ú ýÜÿÆõÂþ¥.j üµÈóî ö¢ÂîÂ ýÂ i 1 á÷ ¥ ýû¤ ©¥¤ ßþÂµÈ : value(S[i, j])ëê üµÈóî ¤¢ ¤ ßþÂ¡ : last(S[i, j])



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KnapSack-4 (M, Weight)1 N ← length[Weight]2 for i← 1 to N3 do for j ← 0 to M4 do V alue(S[i, j])← 05 for i← 1 to N6 do for j ← 1 to M −Wi

7 do k ← j + Wi

8 if j = 0 or V alue(S[i− 1, j]) > 09 then . we may be able to fill S[i, k]

10 if V alue(S[i− 1, k]) < V alue(S[i− 1, j]) + Ci

11 then last(S[i, k])← i12 V alue(S[i, k])← V alue(S[i− 1, j]) + Ci

13 elseS[i, k]← S[i− 1, k]



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:´ª÷ öüõ Ã÷ ¤ ,Ýª ù¢Âî Û ÅþÂõ Âð

Print-One-Result (i, j)1 if j = 02 then return3 k ← last(S[i, j]) . k has to be > 04 Print k5 Print-One-Result (k − 1, j −Wk)


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öüÞ÷ ¤ ùÀª ù¢Ôµ¨ ýû¤ üóø .¢Âî Û Ýû þ¤ ×þ öüõ ¤ ÜÿÆõ ßþ?Â .¢¤ø ´¨¢.´¨ ©¥¤ öÞû S[0..M ] ýþ¤

KnapSack-4 (M, Weight)1 N ← length[Weight]2 for j ← 0 to M3 do S[j]← 04 S[W1]← C1

5 for i← 2 to N6 do for k ← M downto Wi

7 do j ← k −Wi

8 if S[k] < S[j] + Ci

9 then S[k]← S[j] + Ci



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öüÞ÷ ¤ ùÀª ù¢Ôµ¨ ýû¤ üóø .¢Âî Û Ýû þ¤ ×þ öüõ ¤ ÜÿÆõ ßþ?Â .¢¤ø ´¨¢ö¢ÂîÂ ýÂ ´¨ ßØÞõ ,Àª j ýóî ö¢ÂîÂ ýÂ ¤ ßþÂ¡ k Âð îßþ ýÂ.Àª ùÀª ù¢Ôµ¨ k ¤ ¥ Ýû j −Wk ýú



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Â f õî Ci > 0 ,ni <∞ :Ý¹ ´ó ¤¢ Âð) S[i− 1, j] ýÜÿÆõÂþ¥ ¨ ¥ S[i, j] ýÜÿÆõ Âþ¥ ´ó ßþ ¤¢ú ¤¢ Âð) S[i− 1, j −Wi] þ ø ,(Àª ùÀÈ÷ ù¢Ôµ¨ i á÷ ¤ f ¬ ö ýúÇ ú ¤¢ Âð) S[i, j −Wi] ¥ þ (Àª ùÀª ù¢Ôµ¨ i á÷ ¤ ¥ ¢Àä ×þ ÍÖê¢õ ¤ ßþ ¥ üêî ¢Àã îö ¯Âª ,Àª ùÀª ù¢Ôµ¨ i á÷ ¤ ¥ ¢Àä ×þ ¥.´¨ ßþÂµú Àû¢ ´¨¢ ¤ ©¥¤ ßþÂÇ î üµó .Àþüõ ´¨¢ (Àª.Àª ú S[i, j −Wi] ¢¤À÷ üõøÃó ÷Ô¨bµõ.¢¤À÷ ýÀã ø¢ ýþ Ûù¤ ÜÿÆõ Å



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ýÀã¨ ýþ Û

i ¤ î i 1 ýû¤ ¥ j üµÈ óî ö¢Âî Â ýÂ ßþÂ ©¥¤ S[i, j, k].Àª ùÀª ù¢Ôµ¨ ¢Àä k Â·îÀ


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.¢È÷ Â f õî üµÈóî ´¨ ßØÞõ ßþÂ©¥¤ ö¢Âî À ýÂ.Ýþ¢Âðüõ ©¥¤ ßþÂï¤Ã ñ±÷¢ ô N ÂÎ¨ ¤¢ ¤¬ ßþ ¤¢


ce.sharif.educe.sharif.edu/courses/92-93/2/ce354-1/resources/...ûÝµþ¤ Úó Û Ü½ ø ü ÂÏ...

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