Question

I have multiple (N) nested loops as follows:

int k = 0;
for (int i1 = 0; i1 < n; i1++)
{
  for (int i2 = 0; i2 <= i1; i2++)
  {
    for (int i3 = 0; i3 <= i2; i3++)
    {
       ...
            for (int iN = 0; iN <= i{N-1}; iN++)
            { 
              k++;
              //k = f(i1, ... , iN);
            }
    } 
  } 
}

I need a formula to get k inside the loops based on i1, … , iN.

For N=1: k=f(i1)=i1

For N=2: k=f(i1,i2)=i1*(i1+1)/2+i2

Answer 1

In general, you have a nested sum :

sum(sum(sum(1,i3=0..i2),i2=0..i1),i1=0..n-1)

which you can simplify from the inside out using these basic summation formules. The end result will only depend on N as izomorphius commented, as N is the upper limit for i1, which is the upper limit for i2,…

edit: ok, with your edit, I now see you have a different problem in mind. Now it’s a bit more complicated but still no problem.

We will need to split some stuff up for this. I will calculate the value k= f(4,3,1).
At that point in time, we will have performed 4 full i1 loops (i1=0,1,2,3) and are doing the fifth. The k value after 4 full loops of i1(=k_i1) (just before we start the fifth) can be calculated using this function (same kind as before).

k_i1=sum(sum(sum(1,i3=0..i2),i2=0..l),l=0..i1-1) = f1(i1);

Now we start the 5th loop and do the same for the i2 loop. At that point, we will have done 3 full loops of i2 and so we get

k_i2=sum(sum(1,i3=0..l),l=0..i2-1)=f2(i2);

This goes on for all your loops. To get the final value, you have to add each of the fi-functions. k will look like

k=k_i1+k_i2+...

Some minor (+-1) errors in my explanation are possible, but the basic idea is to use the formula to count the full loops.