Time complexity for dependant nested for loop?

Question

Can you explain me how to find time complexity for this?

sum=0;
for(k=1;k<=n;k*=2)
  for(j=1;j<=k;j++)
     sum++;

So, i know the out

Answer 1

I believe you should proceed like the following to formally obtain your algorithm's order of growth complexity, using Mathematics (Sigma Notation):

Answer 2

Just see how many times the inner loop runs:

1 + 2 + 4 + 8 + 16 +...+ n

Note that if n = 32, then this sum = 31 + 32. ~ 2n.
This is because the sum of all the terms except the last term is almost equal to the last term.

Hence the overall complexity = O(n).

EDIT:

The geometric series sum (http://www.mathsisfun.com/algebra/sequences-sums-geometric.html) is of the order of:

(2^(logn) - 1)/(2-1) = n-1.

Answer 3

The outer loop executed log(Base2)n times.so it is O(log(Base2)n).

the inner loop executed k times for each iteration of the outer loop.now in each iteration of the outer loop, k gets incremented to k*2.

so total number of inner loop iterations=1+2+4+8+...+2^(log(Base2)n)
=2^0+2^1+2^2+...+2^log(Base2)n (geometric series)
=2^((log(base2)n+1)-1/(2-1)
=2n-1.
=O(n)
so the inner loop is O(n). So total time complexity=O(n), as O(n+log(base2)n)=O(n).

UPDATE:It is also O(nlogn) because nlogn>>n for large value of n , but it is not asymptotically tight. you can say it is o(nlogn)[Small o] .