This problem is simply a variation on the form

for( int i = 1; i <= N; i *= 2 )
  for( int j = 0; j < i; j++ )
    {
    … some computation using O(1) time …
    }

from the worksheet, which we have discussed in class and in the final review session. It is also very similar to the fast_power using multiplication which takes time proportional to the products of the lengths of the arguments. This form has a time bound of O(N). Adding the third loop is equivalent to making the inner loop of the above program be O(i²) instead of O(i). By the same rationale as before, the overall computation is O(N²), since the final iteration of the outer loop takes longer than all the rest put together.

In more detail for the problem:

for( int i = 1; i <= N; i *= 2 )
  for( int j = 0; j < i; j++ )
    for( int k = 0; k < j; k += 2 )
      {
      … some computation using O(1) time …
      }

It is easy to see that a tight upper bound on the combination of the inner two loops, as a function of i, is ci² where c is an appropriate constant. The outer loop runs I through values 1, 2, 4, 8, …., N/2, N. So an overall bound is

The last term dominates the sum of the other terms, which are all smaller powers of 2. Therefore, the overall time bound is

To see further why the inner two loops are bounded by ci², note that the number of steps of the inner loop as a function of j is j/2 + 1 if j is odd or j/2 if j is even. So as a function of i, we have as the number of steps for the inner two loops:

In the present context, i is always even, except for the case i = 1. The sum for i = 1 is 1 and for i > 1 is the usual sum of an arithmetic series:

which is at most i²/8. We can just as well use c = to get our bound of O(N² ).

           N         T(N)   T(N)/(N^2)   T(N)/(N^3)
           1            0 0.000000e+00 0.000000e+00
           2            1 2.500000e-01 1.250000e-01
           4            5 3.125000e-01 7.812500e-02
           8           21 3.281250e-01 4.101562e-02
          16           85 3.320312e-01 2.075195e-02
          32          341 3.330078e-01 1.040649e-02
          64         1365 3.332520e-01 5.207062e-03
         128         5461 3.333130e-01 2.604008e-03
         256        21845 3.333282e-01 1.302063e-03
         512        87381 3.333321e-01 6.510392e-04
        1024       349525 3.333330e-01 3.255205e-04
        2048      1398101 3.333333e-01 1.627604e-04
        4096      5592405 3.333333e-01 8.138020e-05
        8192     22369621 3.333333e-01 4.069010e-05
       16384     89478485 3.333333e-01 2.034505e-05
       32768    357913941 3.333333e-01 1.017253e-05
       65536   1431655765 3.333333e-01 5.086263e-06