Optimal binary search tree dynamic programming

1. Optimal Binary Search Tree
Dr. P. Subathra
Prof/ IT
KAMARAJ College of Engg. & Tech
(AUTONOMOUS)
Madurai
Tamil Nadu
India

2. • If probabilities of searching for elements of a set are
known—e.g., from accumulated data about past
searches—it is natural to pose a question about an
optimal binary search tree for which the average
number of comparisons in a search is the smallest
possible.
• we limit our discussion to minimizing the average
number of comparisons in a successful search.
• The method can be extended to include unsuccessful
searches as well.
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India
2

3. • The total number of binary search trees with n
keys is equal to the nth Catalan number
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India
3

4. Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India
4

5. OBST CREATION
(j-i)=0
Item 1 2 3 4
Key 10 20 30 40
Freq 4 2 6 3
i j 0 1 2 3 4
0
1
2
3
4
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India
5

6. OBST CREATION
No. of Nodes = 0
(j-i)=0
(0-0) = 0 = C[0,0]
(1-1) = 0 = C[1,1]
(2-2) = 0 = C[2,2]
(3-3) = 0 = C[3,3]
(4-4) = 0 = C[4,4]
Item 1 2 3 4
Key 10 20 30 40
Freq 4 2 6 3
i j 0 1 2 3 4
0 0
1 0
2 0
3 0
4 0
6

7. OBST CREATION
No. of Nodes = 1
(j-i)=1
(1-0) = 1 = C[0,1]
(2-1) = 1 = C[1,2]
(3-2) = 1 = C[2,3]
(4-3) = 1 = C[3,4]
Item 1 2 3 4
Key 10 20 30 40
Freq 4 2 6 3
i j 0 1 2 3 4
0 0
1 0
2 0
3 0
4 0
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India
7

8. OBST CREATION
No. of Nodes = 1
(j-i)=1
(1-0) = 1 = C[0,1] = 41
(2-1) = 1 = C[1,2]
(3-2) = 1 = C[2,3]
(4-3) = 1 = C[3,4]
Item 1 2 3 4
Key 10 20 30 40
Freq 4 2 6 3
i j 0 1 2 3 4
0 0 41
1 0
2 0
3 0
4 0
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India
8

9. OBST CREATION
No. of Nodes = 1
(j-i)=1
(1-0) = 1 = C[0,1] = 41
(2-1) = 1 = C[1,2] = 22
(3-2) = 1 = C[2,3]
(4-3) = 1 = C[3,4]
Item 1 2 3 4
Key 10 20 30 40
Freq 4 2 6 3
i j 0 1 2 3 4
0 0 41
1 0 22
2 0
3 0
4 0
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India
9

10. OBST CREATION
No. of Nodes = 1
(j-i)=1
(1-0) = 1 = C[0,1] = 41
(2-1) = 1 = C[1,2] = 22
(3-2) = 1 = C[2,3] = 63
(4-3) = 1 = C[3,4]
Item 1 2 3 4
Key 10 20 30 40
Freq 4 2 6 3
i j 0 1 2 3 4
0 0 41
1 0 22
2 0 63
3 0
4 0
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India
10

11. OBST CREATION
No. of Nodes = 1
(j-i)=1
(1-0) = 1 = C[0,1] = 41
(2-1) = 1 = C[1,2] = 22
(3-2) = 1 = C[2,3] = 63
(4-3) = 1 = C[3,4] = 34
Item 1 2 3 4
Key 10 20 30 40
Freq 4 2 6 3
i j 0 1 2 3 4
0 0 41
1 0 22
2 0 63
3 0 34
4 0
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India
11

12. OBST CREATION
No. of Nodes = 2
(j-i)=2
(2-0) = 2 = C[0,2]
(3-1) = 2 = C[1,3]
(4-2) = 2 = C[2,4]
Item 1 2 3 4
Key 10 20 30 40
Freq 4 2 6 3
i j 0 1 2 3 4
0 0 41
1 0 22
2 0 63
3 0 34
4 0
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India
12

13. OBST CREATION
No. of Nodes = 2 : (1&2)
C[0,2] : i = 0; j =2; k = 1, 2;
k =1
C[0, 1-1] + C[1, 2]
= min k=2 + (W1+W2)
C[0, 2-1] + C[2, 2]
= min k= 1 (0+2)
k= 2 (0+6)
= 2 + 6 = 81
Item 1 2 3 4
Key 10 20 30 40
Freq (W) 4 2 6 3
i j 0 1 2 3 4
0 0 41 81
1 0 22
2 0 63
3 0 34
4 0
+ (4+2)
13

14. OBST CREATION
No. of Nodes = 2 : (2&3)
C[1,3] : i = 1; j =3; k = 2, 3;
k =2
C[1, 2-1] + C[2, 3]
= min k=3 + (W2+W3)
C[1, 3-1] + C[3, 3]
= min k= 2 (0+6)
k= 3 (2+0)
= 2 + 8 = 103
Item 1 2 3 4
Key 10 20 30 40
Freq
(W)
4 2 6 3
i j 0 1 2 3 4
0 0 41 81
1 0 22 103
2 0 63
3 0 34
4 0
+ (2+6)
14

15. OBST CREATION
No. of Nodes = 2 : (3&4)
C[2,4] : i = 2; j =4; k = 3, 4;
k =3
C[2, 3-1] + C[3, 4]
= min k=4 + (W2+W3)
C[2, 4-1] + C[4, 4]
= min k= 3 (0+3)
k= 4 (6+0)
= 3 + 9 = 123
Item 1 2 3 4
Key 10 20 30 40
Freq (W) 4 2 6 3
i j 0 1 2 3 4
0 0 41 81
1 0 22 103
2 0 63 123
3 0 34
4 0
+ (6+3)
15

16. OBST CREATION
No. of Nodes = 3 :
(j-i) = 3
(3-0) = 3 = C[0,3]
(4-1) = 3 = C[1,4]
Item 1 2 3 4
Key 10 20 30 40
Freq (W) 4 2 6 3
i j 0 1 2 3 4
0 0 41 81
1 0 22 103
2 0 63 123
3 0 34
4 0
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India
16

17. OBST CREATION
No. of Nodes = 3 : (1,2&3)
C[0,3] : i = 0; j =3; k = 1,2,3;
k =1
C[0, 1-1] + C[1, 3]
= min k=2
C[0, 2-1] + C[2, 3]
k=3 + (W1+W2+W3)
C[0, 3-1] + C[3, 3]
= min k= 1 (0+10)
k= 2 (4+6)
k= 3 (8+0)
= 8 + 12 = 203
Item 1 2 3 4
Key 10 20 30 40
Freq (W) 4 2 6 3
i j 0 1 2 3 4
0 0 41 81 203
1 0 22 103
2 0 63 123
3 0 34
4 0
+ (4+2+6)
17

18. OBST CREATION
No. of Nodes = 3 : (2,3 & 4)
C[1,4] : i = 1; j =4; k = 2,3,4;
k =2
C[1, 2-1] + C[2, 4]
= min k=3
C[1, 3-1] + C[3, 4]
k=4 + (W2+W3+W4)
C[1, 4-1] + C[4, 4]
= min k= 2 (0+12)
k= 3 (2+3)
k= 4 (10+0)
= 5 + 11 = 163
Item 1 2 3 4
Key 10 20 30 40
Freq (W) 4 2 6 3
i j 0 1 2 3 4
0 0 41 81 203
1 0 22 103 163
2 0 63 123
3 0 34
4 0
+ (2+6+3)
18

19. OBST CREATION
No. of Nodes = 4 :
(j-i) = 4
(4-0) = 4 = C[0,4]
Item 1 2 3 4
Key 10 20 30 40
Freq (W) 4 2 6 3
i j 0 1 2 3 4
0 0 41 81 203
1 0 22 103 163
2 0 63 123
3 0 34
4 0
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India
19

20. OBST CREATION
No. of Nodes = 4 : (1,2,3 & 4)
C[0,4] : i = 0; j =4; k = 1,2,3,4;
k=1
C[0, 1-1] + C[1, 4]
k =2
C[0, 2-1] + C[2, 4]
= min k=3
C[0, 3-1] + C[3, 4]
k=4 + (W1+W2+W3+W4)
C[0, 4-1] + C[4, 4]
= min k=1 (0+16)
k= 2 (4+12)
k= 3 (8+3)
k= 4 (20+0)
= 11 + 15 = 263
Item 1 2 3 4
Key 10 20 30 40
Freq (W) 4 2 6 3
i j 0 1 2 3 4
0 0 41 81 203 263
1 0 22 103 163
2 0 63 123
3 0 34
4 0
+ (4+2+6+3)
20

21. OBST CREATION
Item 1 2 3 4
Key 10 20 30 40
Freq (W) 4 2 6 3
i j 0 1 2 3 4
0 0 41 81 203 263
1 0 22 103 163
2 0 63 123
3 0 34
4 0
r(0, 4)
3
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India
21

22. OBST CREATION
Item 1 2 3 4
Key 10 20 30 40
Freq (W) 4 2 6 3
i j 0 1 2 3 4
0 0 41 81 203 263
1 0 22 103 163
2 0 63 123
3 0 34
4 0
r(0, 4)
3
r(left, root-1) r(root, right)
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India
22

23. OBST CREATION
Item 1 2 3 4
Key 10 20 30 40
Freq (W) 4 2 6 3
i j 0 1 2 3 4
0 0 41 81 203 263
1 0 22 103 163
2 0 63 123
3 0 34
4 0
r(0, 4)
r(0,2) r(3,4)
3
r(left, root-1) r(root, right)
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India
23

24. OBST CREATION
Item 1 2 3 4
Key 10 20 30 40
Freq (W) 4 2 6 3
i j 0 1 2 3 4
0 0 41 81 203 263
1 0 22 103 163
2 0 63 123
3 0 34
4 0
r(0, 4)
r(0,2) r(3,4)
3
1 4
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India
24

25. OBST CREATION
Item 1 2 3 4
Key 10 20 30 40
Freq (W) 4 2 6 3
i j 0 1 2 3 4
0 0 41 81 203 263
1 0 22 103 163
2 0 63 123
3 0 34
4 0
r(0, 4)
r(0,2) r(3,4)
3
1 4
r(left, root-1) r(root, right)
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India
25

26. OBST CREATION
Item 1 2 3 4
Key 10 20 30 40
Freq (W) 4 2 6 3
i j 0 1 2 3 4
0 0 41 81 203 263
1 0 22 103 163
2 0 63 123
3 0 34
4 0
r(0, 4)
r(0,2) r(3,4)
3
1 4
r(0,0) r(1,2)
r(left, root-1) r(root, right)
2
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India
26

27. OBST CREATION
Item 1 2 3 4
Key 10 20 30 40
Freq (W) 4 2 6 3
i j 0 1 2 3 4
0 0 41 81 203 263
1 0 22 103 163
2 0 63 123
3 0 34
4 0
r(0, 4)
r(0,2) r(3,4)
3
1 4
r(0,0) r(1,2)
r(left, root-1) r(root, right)
2
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India
27

28. OBST CREATION
Item 1 2 3 4
Key 10 20 30 40
Freq (W) 4 2 6 3
i j 0 1 2 3 4
0 0 41 81 203 263
1 0 22 103 163
2 0 63 123
3 0 34
4 0
r(0, 4)
r(0,2) r(3,4)
3
1 4
r(0,0) r(1,2)
r(left, root-1) r(root, right)
2
r(1,1) r(2,2) Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India
28

29. OBST CREATION
Item 1 2 3 4
Key 10 20 30 40
Freq (W) 4 2 6 3
i j 0 1 2 3 4
0 0 41 81 203 263
1 0 22 103 163
2 0 63 123
3 0 34
4 0
r(0, 4)
r(0,2) r(3,4)
3
1 4
r(0,0) r(1,2)
r(left, root-1) r(root, right)
2
r(1,1) r(2,2) Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India
29

30. OBST CREATION
Item 1 2 3 4
Key 10 20 30 40
Freq (W) 4 2 6 3
i j 0 1 2 3 4
0 0 41 81 203 263
1 0 22 103 163
2 0 63 123
3 0 34
4 0
r(0, 4)
r(0,2) r(3,4)
3
1 4
r(0,0) r(1,2)
r(left, root-1) r(root, right)
2
r(1,1) r(2,2) Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India
30

31. OBST CREATION
Item 1 2 3 4
Key 10 20 30 40
Freq (W) 4 2 6 3
i j 0 1 2 3 4
0 0 41 81 203 263
1 0 22 103 163
2 0 63 123
3 0 34
4 0
r(0, 4)
r(0,2) r(3,4)
3
1 4
r(0,0) r(1,2)
r(left, root-1) r(root, right)
2
r(1,1) r(2,2)
r(3,3) r(4,4)
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India
31

32. OBST CREATION
Item 1 2 3 4
Key 10 20 30 40
Freq (W) 4 2 6 3
i j 0 1 2 3 4
0 0 41 81 203 263
1 0 22 103 163
2 0 63 123
3 0 34
4 0
r(0, 4)
r(0,2) r(3,4)
3
1 4
r(0,0) r(1,2)
r(left, root-1) r(root, right)
2
r(1,1) r(2,2)
r(3,3) r(4,4)
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India
32

33. OBST CREATION
Item 1 2 3 4
Key 10 20 30 40
Freq (W) 4 2 6 3
i j 0 1 2 3 4
0 0 41 81 203 263
1 0 22 103 163
2 0 63 123
3 0 34
4 0
3
1 4
2
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India
33

34. OBST CREATION
Item 1 2 3 4
Key 10 20 30 40
Freq (W) 4 2 6 3
i j 0 1 2 3 4
0 0 41 81 203 263
1 0 22 103 163
2 0 63 123
3 0 34
4 0
3
1 4
2
Dr. P. Subathra, KAMARAJ College of Engg &
Tech (AUTONOMOUS), Madurai,
Tamil Nadu, India
34

35. END…!!!