アルゴリズムイントロダクション第2章

2010 CS

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Copyright© 2010 tniky1 All rights reserved. Page 2

2010 CS

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- Θ


2010 CS


2010 CS

j
1
1 2 3 4 5 6
2 5 4 6 1 3
i

length[A]

procedure Insertion-Sort(A)
for j ← 2 to length[A]
do key ← A[j]
i←j−1
while i > 0 and A[i] > key
do A[i+1] ← A[i]
i←i−1
A[i+1] ← key

2010 CS

(1)

(2)


2010 CS

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-  A[1...j-1] →

j

2 5 4 6 1 3

A[1...j-1]


2010 CS

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( )


2010 CS

j

2 5 4 6 1 3

A[1...j-1]

for j ← 2 to length[A] J=2
do key ← A[j] A[1]
A[1..j-1]
i←j−1
while i > 0 and A[i] > key A[j]
A[j-1],A[j-2]...
do A[i+1] ← A[i] A[j]
i←i−1 (A[1..j-1] )
A[i+1] ← key

2010 CS

j=n+1

1 2 3 4 5 6

A[1...j-1]=A[1..n]

for j ← 2 to length[A] j=n+1
do key ← A[j]
A[1..j-1]=A[1..n] !
i←j−1
do A[i+1] ← A[i]
i←i−1
A[i+1] ← key

2010 CS

(1)

(2)


2010 CS

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-  ( )
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-  ( DB
DB )
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2010 CS

j
1 2 3 4 5 6
2 5 4 6 1 3
i

n

for j ← 2 to length[A] n
do key ← A[j] n-1
i←j−1 n-1
do A[i+1] ← A[i]
i←i−1 tj
A[i+1] ← key
n-1

( )

2010 CS

for j ← 2 to length[A] C1 n
do key ← A[j] C2 n-1
i←j−1 C3 n-1
while i > 0 and A[i] > key C4
do A[i+1] ← A[i] C5
i←i−1 C6
A[i+1] ← key
C7 n-1

T(n)


2010 CS

T(n)

( )
j
1 2 3 4 5 6
[ ] 5 6 4 3 2 1
i=0 i
tj=i


2010 CS

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2010 CS


2010 CS

A 2 4 5 7 1 2 3 6
p
q
r

i
j
1 2 3 4 5
1 2 3 4 5
L
2 4 5 7 ∞ R
1 2 3 6 ∞

n1
n2
k

A 1 2 2 3 4 2 3 6
p
q
r


2010 CS

 
-  A L R k−p

-  L[i] R[j] L R A

i
j
1 2 3 4 5
1 2 3 4 5
L
2 4 5 7 ∞ R
1 2 3 6 ∞

k

A 1 2 2 3 4 2 3 6
p
r


2010 CS

i
j

1 2 3 4 5
1 2 3 4 5
L
2 4 5 7 ∞ R
1 2 3 6 ∞

k

A 1 2 2 3 4 2 3 6
p
r

k=p
A[p..k-1]
i=j=1 L,R

L[i]<=R[j]
L[i] A
L[i] A[k]
i k L[i],R[j]


2010 CS

i
j

1 2 3 4 5
1 2 3 4 5
L
2 4 5 7 ∞ R
1 2 3 6 ∞

k

A 1 2 2 3 4 2 3 6
p
r

k=r+1
A[p..k-1]=A[p..r] !

2 n
(n1+n2 ) : Θ(n)


2010 CS

MERGE


2010 CS

A p r
p r

A
/* ( ) */


2010 CS

-  a 1/b

-  If n <= c
-  D(n):
-  C(n):


2010 CS

T(n) = aT(n/b) + D(n) + C(n)

2 Merge Θ(n)
1/2
a = b = 2
D(n) = Θ(1)

T(n) = 2T(n/2) + Θ(1) + Θ(n) if n > 1
=Θ(1) if n = 1


2010 CS

cn( )


2010 CS

log2n


2010 CS

  OK

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アルゴリズムイントロダクション第2章

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