An interesting time complexity question in C++

Time complexity can be defined as the time required by the algorithm to run its average case.

Let's see and calculate the time complexity of some of the basic functions.

Method

void counter(int n){
   for(int i = 0 ; i < n ; i++){
      for(int j = 1 ; j<n ; j += i ){
         cout<<i<<” ”<<j;
      }
      cout<<endl;
   }
}

The above method will run n/I times for all values of i. i.e. n times for the first iteration and 1 time for the last iteration.

According to this, the total time complexity is

(n/1 + n/2 + n/3 + …. + n/n)
= n (1/1 + ½ + ⅓ + …. 1/n)

Now the value of (1/1 + ½ + ⅓ + …. 1/n) is equal to O(log n).

The time complexity of this code is O(nlogn).

Method

void counter(n){
   int i , j ;
   for(int i = 1 ; i <= n ; i++){
      for(j = 1; j <= log(i) ; j++){
         cout<<i<<” ”<<j;
      }
   }
}

The total complexity of the function is O(log 1) + O(log 2) + O(log 3) + …. + O(log n) which is O(log n!).