For Education

1. DESIGN ANALYSIS AND
ALGORITHM
TOPIC : OPTIMAL BINARY SEARCH TREE

2. BINARY TREE
 Each parent node has atmost two child nodes.
 It has parent node and 2 leaf nodes
 This is called a binary tree.
 Eg:
2
7 5
3 6 3 8

3. BINARY SEARCH TREE
 EXAMPLE :
8
3 10
1 6
14

4. BINARY SEARCH TREE ( CONT..)
 Left child values should be less than the parent
node or Root node.
 Right child values should be greater than Root
Node.
 This is called Binary search tree.

5.  For the purpose of searching we have a different
types of searches.
 Linear search
 Binary search
 Binary tree data structure.

6. BINARY TREE DATA STRUCTURE
 Binary tree is mainly used for searching purpose
also and it is very easy to identify the elements.
 For searching an element in binary tree,the
maximum time is taken, when the element is
present in a leaf.
 The height of the binary tree is log n.
 In order to rectify the cost of search we are using
optimal binary search tree.

7. OPTIMAL BINARY SEARCH TREE
 The arrangement of elements in the form of a
binary data structure such that the cost of searching
the elements is minimum.
 This is called optimal binary search tree.
 To reduce the cost of searching we use optimal
binary search tree.
 The word “optimal” means best so , to search an
element we use the best way that is optimal binary
search tree.

8.  Eg: Consider a set of elements
10,20,30
Cost of searching is 1+2+3=6 ( maximum)
1
2
3
10
20
30

9. OPTIMAL BINARY TREE:
Cost of searching:
1+2+2 = 5 ( minimum )
1
2 2
20
10 30

10. 0 4 81 203 263
0 2 103 163
0 6 123
0 3
0
0 1 2 3 4
0
1
2
3
4
i
j

11.  Step 1: Consider the cost of reaching the same node
 c[0,0]=0
 c[1,1]=0
 c[2,2]=0
 c[3,3]=0
 C[4,4]=0
Step2: Consider the cost of reaching one node
• c[0,1]=4
• c[1,2]=2
• c[2,3]=6
• c[3,4]=3

12.  Step 3: Consider cost of reaching 2 nodes distance
 c[0,2], c[1,3] , c[2,4]
 c[i,j]= min{ c[i,k-1] + c[k,j] + w[i,j]}
 Where k=1 ( rootnode)
 c[0,2]= min { c[0,0]+c[1,2]+w[0,2],c[0,1]+c[2,2]+w[0,2]}
 =min{ 0+2+6, 4+0+6}
 =min{8,10}
 C[0,2]=81
 c[1,3]= min { c[1,1]+c[2,3]+w[1,3],c[1,2]+c[3,3]+w[1,3]}
 =min{ 0+6+8,2+0+8}
 =min{14,10}
 C[1,3]=103

13.  c[2,4]= min { c[2,2]+c[3,4]+w[2,4],c[2,3]+c[4,4]+w[2,4]}
 =min{ 0+3+9, 6+0+9}
 =min{12,15}
 C[2,4]=123
 Step 4: Consider the cost of reaching 3 nodes.
 c[0,3]= min {
c[0,0]+c[1,3]+w[0,3],c[0,1]+c[2,3]+w[0,3],c[0,2]+c[3,3]+12}
 =min{ 0+10+12,4+6+12,8+0+12}
 =min{22,22,20}
 C[0,3]=203

14.  c[1,4]= min {
c[1,1]+c[2,4]+w[1,4],c[1,2]+c[3,4]+w[1,4],c[1,3]+
 c[4,4]+w[1,4]}
 =min{ 0+12+11,2+3+11,10+0+11}
 =min{23,16,21}
 C[1,4]=163
 Step 5: Consider the cost of reaching 4 nodes.
 c[0,4]= min {
c[0,0]+c[1,4]+w[0,4],c[0,1]+c[2,4]+w[0,4],c[0,2]+c[3,
4]+w[0,4],c[0,3]+c[4,4]+w[0,4]}
 =min{ 31,31,26,35}
 =min{22,22,20}
 C[0,4]=263

15. Optimal binary search tree:
10 40
20
30

16. THANK YOU