# How much times command executed? Looking for mistake

I have following piece of code:

int sum = 0;
for (int i = 1; i <= N; i++)
for (int j = 1; j <= N; j++)
for (int k = 1; k <= N; k = k*2)
for (int h = 1; h <= k; h++)
sum++;

So I'v calculated how much does each cycle executes and then whole script, and I think I might be wrong for the last one.

1. N
2. N
3.  1 + log2N( means log N to the base 2)
4. N

So the total execution amount of inner cycle would be N^3 * (1 + log2N).

Am I right? How I can transform this statement?

UPDATE 1

I have another solution which seems monstrous:

1. N
2. N
3. LOG2(N) + 1
4. 2^(LOG2(2N)) - 1

So total cycles amount would be n^2 * (LOG2(N) + 1) * 2^(LOG2(2N)) - 1.

Which transforms to n^2 order of growth.

UPDATE 2

I wrote simple test app to check my assumption.

it seems that third cycle is already calculated for some reason is fourth one. At least this is the result of test app. I threw away (LOG2(N) + 1).

As the result of several transformations I have following equation for total amount of sum++ calls:

N*N*(2*N - 1) = 2 * N^3 - N^2 ~ N^3
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## 1 Answer

The first two loops are not interesting - they just give N^2 multiplier.

The fourth loop inside of third loop gives a geometrical progression: 1,2,4,8,...

Totally it will increase sum by 2M-1 where M is the largest power of 2 less than N.

So totally: N^2*(2*M-1)

In terms of big-O that's O(N^3).

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So what would be your verdict? May I consider, that this would be N^3 or N^2? From your opinion I would think that this would be N^3. Because, in general I would need the worst case solution. And worst case is when M == N, another words M is power of 2. – Jevgeni Smirnov Feb 13 '13 at 12:55
It could make sense to make more clear that you want big-O estimation, not exact answer ;), yes, it will be O(N^3). – maxim1000 Feb 13 '13 at 13:00