Java Big O notation of 3 nested loops of log(n)

Question

What would the Big O notation be for the following nested loops?

     for (int i = n; i > 0; i = i / 2){
        for (int j = n; j > 0; j = j / 2){
           for (int k = n; k > 0; k = k / 2){
              count++;
           }
        }
     }

My thoughts are: each loop is O(log2(n)) so is it as simple as multiply

O(log2(n)) * O(log2(n)) * O(log2(n))  =  O(log2(n)^3)

My assumption also would be `O(log2(n)^3)`. – Subhrajyoti Majumder Jan 23 '13 at 06:31 — Subhrajyoti Majumder, Jan 23 '13 at 06:31

score 11 · Accepted Answer · answered Jan 23 '13 at 06:29

11

Yes, that is correct.

One way to figure out the big-O complexity of nested loops whose bounds do not immediately depend on one another is to work from the inside out. The innermost loop does O(log n) work. The second loop runs O(log n) times and does O(log n) work each time, so it does O(log² n) work. Finally, the outmost loop runs O(log n) times and does O(log² n) work on each iteration, so the total work done is O(log³ n).

Hope this helps!

answered Jan 23 '13 at 06:29

templatetypedef

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what is the correct notation? O(log2(n) ^ 3) or the way you have it? or are they both acceptable? – Kailua Bum Jan 23 '13 at 06:31
I've seen this written both ways. I personally like log^3 n in the style of sin^2 x, though go with whatever convention is used in context. – templatetypedef Jan 23 '13 at 06:32
The notation for `log² n` could be confused for `log log n` – Peter Lawrey Jan 23 '13 at 08:37

Subhrajyoti Majumder · Answer 2 · 2013-01-23T06:44:21.467

Yes you are right.

Easy way to calculate -

for(int i=0; i<n;i++){ // n times 
    for(int j=0; j<n;j++){ // n times
    }
}

This example of simple nested loop. Here Big-O of each loop O(n) and it is nested so typically O(n * n) which is O(n^2) actual Big-O. And in your case -

for (int i = n; i > 0; i = i / 2){ // log(n)
     for (int j = n; j > 0; j = j / 2){ // log(n)
         for (int k = n; k > 0; k = k / 2){ // log(n)
           count++;
         }
     }
}

Which is in nested loop where each loop Big-O is O(log(n)) so all together complexity would be O(log(n)^3)

score 0 · Answer 3 · answered Apr 05 '14 at 00:49

0

Indeed, your assumption is correct. You can show it methodically like the following:

enter image description here

answered Apr 05 '14 at 00:49

Mohamed Ennahdi El Idrissi

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Java Big O notation of 3 nested loops of log(n)

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