Optimal Binary Search tree
Optimal binary search trees
e.g. binary search trees for 3, 7, 9, 12;
3
7
3
7
9
7
12
3
12
9
9
9
12
3
7
12
(a)
(b)
(c)
Algorithm Analysis and Design CS 007 BE CS 7th Semester
(d)
2
Optimal binary search trees
n identifiers : a1 <a2 <a3 <…< an
Pi, 1in : the probability that ai is searched.
Qi, 0in : the probability that x is searched
where ai < x < ai+1 (a0=-, an+1=).
n
n
 P  Q
i 1
i
i 1
i
1
Algorithm Analysis and Design CS 007 BE CS 7th Semester
3
10
5
14
4
E0
8
E1
E2
11
E3
E4
E7
12
E5

Identifiers : 4, 5, 8, 10,
11, 12, 14
Internal node : successful
search, Pi
External node :
unsuccessful search, Qi
E6
The expected cost of a binary tree:
n
n
 P  level(a )   Q  (level(E )  1)
n 1

i
i
n 0
i
i
The level of the root : 1
Algorithm Analysis and Design CS 007 BE CS 7th Semester
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The dynamic programming
approach
Let C(i, j) denote the cost of an optimal binary
search tree containing ai,…,aj .
The cost of the optimal binary search tree with ak
as its root :
k 1
n


 

C(1, n)  min Pk  Q0   Pi  Qi   C1, k  1  Q k   Pi  Qi   Ck  1, n  
1 k  n
i 1
i  k 1

 


P1...Pk-1
Q0...Qk-1
a1...ak-1
ak
Pk+1...Pn
Qk...Qn
ak+1...an
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C(1,k-1)
C(k+1,n)
5
General formula
k 1



C(i, j)  min  Pk  Qi-1   Pm  Q m   Ci, k  1
i k  j
m i



j


 Q k   Pm  Q m   Ck  1, j 
m  k 1


j


 min Ci, k  1  Ck  1, j  Qi-1   Pm  Q m 
i k  j
m i


P1...Pk-1
Q0...Qk-1
a1...ak-1
ak
Pk+1...Pn
Qk...Qn
ak+1...an
Algorithm Analysis
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C(1,k-1)
C(k+1,n)
6
Computation relationships of
subtrees
e.g. n=4
C(1,4)
C(1,3)
C(1,2)
C(2,4)
C(2,3)
C(3,4)
Time complexity : O(n3)
when j-i=m, there are (n-m) C(i, j)’s to compute.
Each C(i, j) with j-i=m can be3computed in O(m) time.
O(
 m(n  m))  O(n )
1Algorithm
m n Analysis and Design CS 007 BE CS 7th Semester
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