Algorithm Analysis, Time Complexity of algorithm

Question

m=1;
for(i=1;i<=n;i++){
    m=m*2;
    for(j=1;j<=m;j++){
        do something that is O(1)
    }
}

What will be time complexity of the above code ?? Please tell me how to solve these types of problem.

score 5 · Answer 1 · answered Sep 23 '13 at 14:06

5

I prefer to look at these problems from the inside out. Removing the m, we have:

for(i=1;i<=n;i++){
    for(j=1;j<=2^i;j++){
        do something that is O(1)
    }
}

Or:

for(i=1;i<=n;i++){
    O(2^i)
}

So overall: sum_1^n O(2^i)=O(2^(n+1))=O(2^n).

answered Sep 23 '13 at 14:06

Teepeemm

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Ari · Answer 2 · 2013-09-23T19:04:42.177

4

Inner loop will iterate 1 time, then 2 times, then ..., then 2^n times. So we have 1 + 2 + 4 + ... + 2^n = 2^(n + 1) - 1 = O(2^n) iterations of inner loop.

One iteration of inner loop has constant complexity, so summed_inner_loop_complexity = O(2^n).

Whole complexity is O(2^n).

edited Sep 23 '13 at 19:04

answered Sep 23 '13 at 13:46

Ari

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No. You can multiply by `n`, or get the summed inner loop complexity (better), but both means you're over counting. The answer should be `O(2^n)`. – Teepeemm Sep 23 '13 at 14:01
I think it should be O(2^n) as Teepeemm has explained. – Ashok Naval Sep 23 '13 at 14:49

score 0 · Answer 3 · answered Jun 06 '14 at 20:30

0

Formally and methodically, you can use Sigma notation:

enter image description here

answered Jun 06 '14 at 20:30

Mohamed Ennahdi El Idrissi

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Algorithm Analysis, Time Complexity of algorithm

3 Answers3