(i)Find T(n), the time complexity (as operations count) in the worst case? (ii)Express the growth rate of the function in asymptotic notation in the closest bound possible. (iii)Prove that T(n) is Big O (g(n)) by the definition of Big O (iv)Prove that T(n) is (g(n)) by using limits Solution Operations Times Sum=0 C1 1 for(int i=0;i<n;i++) C2 n for(int j=0;j<n;j++) C3 n for(int k=0;k<n;k*2){ C4 Log n sum=i+j+k; C5 Log n } i) fT(n) =n 2 log n 2) The growth rate is O(n 2 log n) 3)T(n) is Big O(n 2 log n) Operations Times i=1; C1 1 While(i<=n){ C2 n J=I; C3 n While(j<=n){ C4 n^2 System.out.print(j); C5 n^2 J++; C6 n^2 } I++; C7 n } 1. T(n) is n^2 2.O(n^2) .

1. (i)Find T(n), the time complexity (as operations count) in the worst case?
(ii)Express the growth rate of the function in asymptotic notation in the closest bound possible.
(iii)Prove that T(n) is Big O (g(n)) by the definition of Big O
(iv)Prove that T(n) is (g(n)) by using limits
Solution
Operations
Times
Sum=0
C1
1
for(int i=0;i<n;i++)
C2
n
for(int j=0;j<n;j++)
C3
n
for(int k=0;k<n;k*2){
C4

2. Log n
sum=i+j+k;
C5
Log n
}
i) fT(n) =n 2
log n
2) The growth rate is O(n 2
log n)
3)T(n) is Big O(n 2
log n)
Operations
Times
i=1;
C1
1
While(i<=n){
C2
n
J=I;
C3
n
While(j<=n){
C4
n^2
System.out.print(j);

3. C5
n^2
J++;
C6
n^2
}
I++;
C7
n
}
1. T(n) is n^2
2.O(n^2)
Operations Times
Sum=0 C1 1
for(int i=0;i<n;i++) C2 n
for(int j=0;j<n;j++) C3 n
for(int k=0;k<n;k*2){ C4 Log n
sum=i+j+k; C5 Log n
}