Question

Big-0 Notation

Asked by: sidd_barai

1.  In the definition of Big-O, why is the "for N >= n0" needed?
2. Since Big-O notation is a mathematical tool for functions like f(N) or g(N), how is
    it applicable to algorithm analysis?

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Asked On
2009-09-11 at 22:26:36ID24726474
Tags

Big -0

Topics

Algorithms

,

Kernel And Operating System Specific Programming

,

Editors IDEs

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Answers

 

by: ozoPosted on 2009-09-11 at 22:34:51ID: 25315252

We only care what happens when N gets large, so we can can ignore things that only happen at small N

We can compare how different algorithms are affected as N gets large, independently of  details of implementation.

 

by: daryalPosted on 2009-09-12 at 00:55:25ID: 25315559

 

by: JoseParrotPosted on 2009-09-14 at 16:38:19ID: 25330234

Big O notation is useful for algorithm analysis to determine the time needed for its execution and mainly to preview how much grows the time execution, in function of n, being n the size of the input of the problem.

For exemple, in ALG 1 below, the algorithm spends n times the time needed to draw a line. Then the analysis of such algorithm shows that it is in O(n).

In ALG 2, the algorithm spends n times the time needed to draw two lines. In terms of timing, it will probably spend twice the time as in ALG 1. The analysis of ALG 2 shows that it is in O(n) too, because the time grows proportinaly to n, not to the time to draw a line.

In ALG 3 we have a quadratic growth, say, O(n²), say the time grows proportionaly to the power of two of n.

Please note, Big O notation is applied also for memory space.

Jose

// ALG 1
for i=1 to n
{
   draw a line
}
--------------------------
// ALG 2
for i=1 to n
{
   draw two lines
}
--------------------------
// ALG 3
for i = 1 to n
{
   for j = 1 to n
   {
       draw a line
   }
}

                                              
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by: tormorkPosted on 2009-11-07 at 17:11:41ID: 25768914

Often a good algorithm uses more time for preparation, and will therefore be able to outrun a lesser algorithm when the dataset is large. But if the dataset is small the lesser algorithm can finish while the better algorithm is still preparing.

But when you start working with O notation the aim is to get an algorithm to run faster when the dataset is large, while it is not interesting if it take 1 millisek or 2 millisek to run with a small dataset, it is important to run a large set in 1 hour and not 2.

The problem in the wording "for large N" is that you can exploit it say you want to sort a list of integers, mathematically a large N could be interpreted as the number of integers all computers can store and therefore you can claim that any sorting algorithm will run in O(1) time for really large N :) don't do that!

If you have two algorithms that run: A: N*N  (ie. O(N*N) and B: N+10 (i.e O(N)) and plot them in a graph you will soon see what the point is:

The runtimes will be:
1 element   A: 1  B: 11
2 elements A: 4  B: 12
3 elements A: 9  B: 13
4 elements A:16 B: 14
5 elements A:25 B: 15

In this case the "large N" is 4 where the O(N) start running faster than the O(N*N), but when there are 1-3 elements O(N*N) is better.

20120131-EE-VQP-002

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