Runtime of nested loops

Question

What is the asymptotic runtime of fthe ollowing piece of code in terms of number of updates to S in worst case.

Void foo( int n)  
{  

    int S = 0;  

   for ( i=1 ; i<= n ; i++)  
    {  

           for( j = 1 ; j<= i*i ; j++)  
               {  
                  if(j % i == 0)  
                     {  
                       for( k=1 ; k<=j ; k++)  
                       s++;  
                     }  
               }  
     }  
}   

My way :

1. Outer for loop will execute 'n' time.

2. Middle for loop will execute 'n^3' times. Because j is reaching to i*i.
   So if n =10 ,i will move from 1 to 10. j will execute 1*1 + 2*2 + 3*3 + ..... + 10*10. i.e. 1^2 + 2^2 + ... + n^2 = maximum term is n^3.

3. Inner for loop will execute maximum 'n' times. Suppose n = 10 , and i reaches to 10 i.e. i = 10 , so j can reach 1 to 10 * 10. In this maximum 10 times i.e. n times 'if loop' condition will be fulfilled.

   so total n * n^3 * n = Theta ( n^5 )

But answer is given as Theta ( n^4 ) and explained in totally mathematical terms ( which I didnt get )

Did I make any mistake ?

Answer 1

You can get a better bound by taking into account the fact that the innermost loop only runs $1/i$ of the times. The number of times that the body of the $j$-loop is executed is $$ \sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6} = \Theta(n^3), $$ as you notice correctly. The innermost loop is run whenever $j$ is a multiple of $i$, that is, for the values $ti$ for $t=1,\ldots,i$. Therefore the $k$-loop is executed this many times: $$ \sum_{i=1}^n \sum_{t=1}^i ti = \sum_{i=1}^n i \sum_{t=1}^i t = \sum_{i=1}^n i \frac{i(i+1)}{2} = \frac{n(n+1)(n+2)(3n+1)}{24} = \Theta(n^4).$$ In total, the running time is $\Theta(n^4)$. Note that there is no need to calculate the exact formula; for example, we could use the rough lower and upper bounds $$ \sum_{i=1}^n i \sum_{t=1}^i t \geq \sum_{i=n/2}^n (n/2) \sum_{t=n/4}^{n/2} (n/4) = n^4/64, \\ \sum_{i=1}^n i \sum_{t=1}^i t \leq \sum_{i=1}^n n \sum_{t=1}^n n = n^4, $$ which are good enough for our purposes.