This document discusses optimal binary search trees and provides an example problem. It begins with basic definitions of binary search trees and optimal binary search trees. It then shows an example problem with keys 1, 2, 3 and calculates the cost as 17. The document explains how to use dynamic programming to find the optimal binary search tree for keys 10, 12, 16, 21 with frequencies 4, 2, 6, 3. It provides the solution matrix and explains that the minimum cost is 2 with the optimal tree as 10, 12, 16, 21.

1. Design and Analysis of Algorithm(18CS42)
Module 4
Dynamic programming
Lecture Presentation
Divya K S
Dept. of CSE

2. OPTIMAL BINARY SEARCH TREE
• Basic definitions
• Concepts of Optimal Binary Search Tree
• Example Problem
• Assignment

3. BINARY SEARCH TREE
• In A Binary Search Tree, The Value Of All The Nodes In The Left Sub-tree Is Less Than The Value Of The
Root.
• Similarly, Value Of All The Nodes In The Right Sub-tree Is Greater Than Or Equal To The Value Of The
Root.
• This Rule Will Be Recursively Applied To All The Left And Right Sub-trees Of The Root.

4. EXAMPLE

5. KEYS 1, 2,3

6. BST 3BST 2
BST 4
BST 1
BST
5

7. DEPTH=0,HEGHT=1
DEPTH=1,HEIGHT=2
DEPTH=2 HEIGHT=3
HEIGHT*FREQUEN

8. 1*1+2*2+2*6=17

11. Cost: 13

12. • Using dynamic programming find the optimal binary search tree for
the given data.
Keys 10 12 16 21
Frequency 4 2 6 3
0 1 2 3 4
1 0 4
2 0 2
3 0 6
4 0 3
5 0
1 2 3 4
Keys 10 12 16 21
Frequency 4 2 6 3
Solution: Solution matrix

13. Formula:
0 1 2 3 4
1 0 4 8(1)
2 0 2 10(3
)
3 0 6
4 0 3
5 0
C[1,2]=
K=1
C[1,0]+C[2,2]=0+2=2---------
ROOT(1)
K=2
C[1,1]+C[3,2]=4+0=4
+ 6===8
C[2,3]=
I=2
J=3
K=2
C[2,1]+C[3,3]=0+6=6
K=3
C[2,2]+C[4,3]=2+0=2
+8===10

14. 0 1 2 3 4
1 0 4 8(1) 20(3
)
26(3
)
2 0 2 10(3
)
16(3
)
3 0 6 12(3
)
4 0 3
5 0
C[3,4]
k=3
C[3,2]+C[4,4]=0+3=3
i=3,j=4 k=4
C[3,3]+C[5,4]=6+0=6
Frequency=9
=====12ROOT(3)
Minimum cost =2
16
10 21
12 Optimal Binary search
1*6=6
2*4+2*3=14
3*2=6

15. ALGORITHM OPTIMAL BINARY SEARCH TREE

16. ASSIGNMENT