How to find the asymptotic runtime of these nested loops?

Question

i=n; 
while(i>0) {
  k=1;
  for(j=1;j<=n:j+=k)
    k++;
  i=i/2;
}

The while loop has the complexity of $\lg(n)$ the j value of inner loop runs 1,3,6,10,15... increase like 2,3,4,5,...

But how to find the overall complexity ?

From your comments it seems as if you don't know even the basics of algorithm analysis. Please see many questions of the same kind (1, 2, 3) and our reference questions; they should contain everything you need in sufficient detail. — Raphael, Jan 03 '14 at 15:56
I misread the question earlier; the k++ in the inner loop makes things more interesting. Still, please read the questions I linked and present a more elaborate attempt which we can then work off. — Raphael, Jan 03 '14 at 16:00
Please stop undoing edits that improve the question or its formatting. — Raphael, Jan 04 '14 at 19:37
As I've said before, as long as you don't include an elaboration on your thoughts, it remains a duplicate. — Raphael, Jan 05 '14 at 16:48

score 1 · Accepted Answer · answered Jan 03 '14 at 12:56

1

$j$ satisfies the recurrence $j_k = j_{k-1}+k$. Note that another sequence, $s_k = k^2$ satisfies $s_k = s_{k-1} + 2k-1$.

$$j_k = \sum_{i=1}^k i = \frac 1 2\sum_{i=1}^k 2i = \frac 1 2(s_k+k)$$

This should give you enough to find the largest $k$ such that $j_k < n$.

answered Jan 03 '14 at 12:56

Karolis Juodelė

why is the second relation needed can u explain @Karolis – Xax Jan 03 '14 at 13:06
@avi, which relation is the second? Are you asking why bother expressing $j_k$ in terms of $s_k$? – Karolis Juodelė Jan 03 '14 at 14:00
yes ,i cant figure it out .@Karolis – Xax Jan 03 '14 at 14:25
@avi, if the loop was for(k = 1; s_k < n; k++) the complexity would be $O(\sqrt n)$. – Karolis Juodelė Jan 03 '14 at 14:35
total complexity is then O(log n^3/2)@Karolis – Xax Jan 03 '14 at 14:45
@avi, I'm not sure where you put the parentheses in that, but no. You had already decided the total complexity would be $O(k\log n)$ and this isn't that. – Karolis Juodelė Jan 03 '14 at 14:54
So what is the complexity of the segment?@Karolis – Xax Jan 03 '14 at 15:03
@avi Translate the loops to sums and evaluate... – Raphael Jan 03 '14 at 15:43

1 Answers1