Question

find the big oh characterization

input: n  

s<-0  
for i<-1 to n^2 do  
  for j<-1 to i do  
    s<-s+i

Both loops iterate n^2 times. I’m not sure if I should add or multiply the loops. Right now I think the answer is O(n^4).

Answer 1

The outer loops i from 0 to n^2, while the inner one loops j from 1 to i.

Thus, the inner loop has complexity of i (inner loop needs i steps, i is varying). Let us denote the runtime of the inner loop by IL(i)=i.

To get the total runtime, we see that the inner loop is executed n^2 times with varying “parameter” i. Thus, we obtain as total runtime:

 n^2           n^2
----          ----
\     IL(i) = \    i 
/             /
----          ----
 i=1           i=1

That is, you have to sum up all numbers from 1 to n^2. There are many basic textbooks explaining how to evaluate this.

The result is (n^2)(n^2+1)/2, which leads to an overal complexity of O(n^4).