Big-Oh Notation,algorithm-analysis and running time - aagh headache!

Question

Hey folks,
I was reading about the Big-Oh Notation, algorithm-analysis and running time.
Now, it's all probably very nice - if you get to understand it. Which I don't! Gives me headache! lol
And we're having an exam soon. Hmm. Could someone please help to explain it a bit better please?
Oh and by the way, one of the examples mention something about T(N), which I presume is running time for the size of a problem.
So it said:
"? Running time of a loop =
time taken by the body statement * size of the loop
? Example
for ( i = 1; i <= N; i ++)
c = c + 1;
? body statement takes O(1), the size of the loop is N
? So, this loop has running time T(N) = N * O(1) = O (N).
? More precise ?
? i = 1 takes 1 time unit
? Body statement takes 2 time units
? i <= N takes 1 time unit, i ++ takes 1 time unit
? Totally T (N) = 1 + 1 * (N +1) + 2 * N + 1 * N = 4N + 2 = O(N)
init condition body update
Nested loops
? Nested loops: Analyze them inside out
time take by body statement of innermost loop *
product of the size of all the loops
? Example:
line 1 for ( i = 1; i <= N; i ++)
line 2 for ( j = 1; j <= N; j ++)
line 3 k++;
? line 3 is the body statement in the innermost loop (for-j): takes
O(1).
? size of for-j: N
? size of for-i: N
? So the running time for this nested loops: N*N*O(1) = O(N2 )"
Please help me! I don't understand much of this.
Best Wishes!

Answer 1

Ok. I'll have a look about it - but the question I
had was... like...
what exactly is it that takes N steps? How do I know
what takes O(1) steps
and N steps?Depends on what you're doing. Example: adding an item to a linked list of N elements will always take the same effort: attach it to the end. Whether the list already has 0, 10 or 1000 elements doesn't matter. O(1).
If you want to try and find the maximum value in an array of numbers, you need to look at each number in there. So an array with 1 number takes one look, an array with 100 numbers needs 100 looks and an array with N members needs N looks. O(N).
Does the parametres themselves take a step each? How
many steps does the body (as far as I've understood
it, it's c = c + 1; which is the body) take? And how
long time does it take?That body takes only one step. The question is whether you want to account for the repetition or not. If you don't, it's O(1) . There is no N to take into account. If you count in the repetitions too, you end up with an N - the number of times you call the body. So the effort O to run the body once is O(1), the effort to run it N times using the loop is O(N). It really depends on what the actual action is you want to look at.