アルゴリズムイントロダクション 第2章の勉強会資料 下記でPPT版，KEY版も配布してます． https://tanaike.jp

1. 2010 CS

2. 2010 CS


Copyright© 2010 tniky1 All rights reserved. Page 2

3. 2010 CS

-
-

-

- Θ
Copyright© 2010 tniky1 All rights reserved. Page 3

4. 2010 CS
Copyright© 2010 tniky1 All rights reserved. Page 4

5. 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
Copyright© 2010 tniky1 All rights reserved. Page 5

6. 2010 CS
(1)
(2)
Copyright© 2010 tniky1 All rights reserved. Page 6

7. 2010 CS


- A[1...j-1] →
j
2 5 4 6 1 3
A[1...j-1]
Copyright© 2010 tniky1 All rights reserved. Page 7

8. 2010 CS

-
-
-
-
-
-
( )
Copyright© 2010 tniky1 All rights reserved. Page 8

9. 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
Copyright© 2010 tniky1 All rights reserved. Page 9

10. 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
while i > 0 and A[i] > key
do A[i+1] ← A[i]
i←i−1
A[i+1] ← key
Copyright© 2010 tniky1 All rights reserved. Page 10

11. 2010 CS
(1)
(2)
Copyright© 2010 tniky1 All rights reserved. Page 11

12. 2010 CS

-
-
- ( )

- ( DB
DB )
-
Copyright© 2010 tniky1 All rights reserved. Page 12

13. 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
while i > 0 and A[i] > key
do A[i+1] ← A[i]
i←i−1 tj
A[i+1] ← key
n-1
( )
Copyright© 2010 tniky1 All rights reserved. Page 13

14. 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)
Copyright© 2010 tniky1 All rights reserved. Page 14

15. 2010 CS
T(n)
( )
j
1 2 3 4 5 6
[ ] 5 6 4 3 2 1
i=0 i
tj=i
Copyright© 2010 tniky1 All rights reserved. Page 15

16. 2010 CS

-
-

-
-
-
Copyright© 2010 tniky1 All rights reserved. Page 16

17. 2010 CS
Copyright© 2010 tniky1 All rights reserved. Page 17

18. 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
Copyright© 2010 tniky1 All rights reserved. Page 18

19. 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
Copyright© 2010 tniky1 All rights reserved. Page 19

20. 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]
Copyright© 2010 tniky1 All rights reserved. Page 20

21. 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)
Copyright© 2010 tniky1 All rights reserved. Page 21

22. 2010 CS
MERGE
Copyright© 2010 tniky1 All rights reserved. Page 22

23. 2010 CS
A p r
p r
A
/* ( ) */
Copyright© 2010 tniky1 All rights reserved. Page 23

24. 2010 CS
- a 1/b
- If n <= c
- D(n):
- C(n):
Copyright© 2010 tniky1 All rights reserved. Page 24

25. 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
Copyright© 2010 tniky1 All rights reserved. Page 25

26. 2010 CS
cn( )
Copyright© 2010 tniky1 All rights reserved. Page 26

27. 2010 CS
log2n
Copyright© 2010 tniky1 All rights reserved. Page 27

28. 2010 CS
 OK
-
-
Copyright© 2010 tniky1 All rights reserved. Page 28