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Help me review for exam

 
Tonks Gabriel
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Can anyone explain to me the difference between:

For each of the following loops with a method call, determine the overall complexity. As above, assume that method f takes constant time, and that method g takes time linear in the value of its parameter.

1. for (j = 0; j < N; j++) f(j);

2. for (j = 0; j < N; j++) g(j);

3. for (j = 0; j < N; j++) g(k);

I understand why number 1 is O(n), but I don't get it why 2 and 3 are O(n^2). Can you explain it to me in a non-mathematical way. Thanks
 
Stephan van Hulst
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Hi Tonks,

In the second case, you perform g(0), g(1), g(2), ..., g(N). If g performs in linear time, then the loop will take 0 + 1 + 2 + ... + N to complete, or (N^2)/2. The division by 2 has no bearing on the complexity. The complexity class only considers the degree of the polynomial, in this case N^2.

The third case has a wrong answer. The answer is O(N*k), or O(N) if k remains constant. Since k is a constant, the method call will always perform in constant time.
 
Tonks Gabriel
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Hi Stephen,

I thought Linear time is O(n)?


Also, how would you determine the big o notation of the following given:

Example:

8n^3+5000n^2, 5. (m + k)(10kn3 + 50kn), or 2^n+1
 
Stephan van Hulst
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Yes, linear time is O(n). Why do you ask?

As for your examples, the only thing that matters is the degree (the highest power) of the polynomial.

For your three examples, those would be O(n^3), O(mk(n^3) + (k^2)(n^3)) and O(2^n). The second one can be simplified to O(n^3) if m increases less fast that cubic time, and k increases less fast than power 1.5 time.
 
Stephan van Hulst
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Note that your third example is in exponential time, not polynomial time.
 
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