Dumbledore's blog

By Dumbledore, history, 4 years ago, In English,
for (int i = 1; i <= MAXN; i++)
     for (int j = i; j <= MAXN; j += i);


Can anybody explain to me why the time complexity for this code is O(N log N)?

 
 
 
 
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4 years ago, # |
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The time complexity is O(N + N/2 + N/3 + N/4 + ... + N/N) = O(N*(1 + 1/2 + 1/3 + ... + 1/N)).

But 1 + 1/2 + 1/3 + ... 1/N < log(N). (link)

=> O(N + N/2 + N/3 + N/4 + ... + N/N) = O(N*(1 + 1/2 + 1/3 + ... + 1/N)) = O(N*log(N)).

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4 years ago, # |
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4 years ago, # |
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Already answered twice, but still:

For the number of iterations (complexity) you have the sum :

Where Hn is the sum of the first n terms of the Harmonic series. Generally the harmonic series differs by a constant factor from the natural logarithm, more precisely, if n tends towards infinity then the difference Hn - ln(n) tends towards the Euler-Mascheroni constant which has a value of about 0.577, hence we get that the complexity is O(NlnN) = O(NlogN)