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Big O Notation

lisbon_lions

Hi,

Does anyone have a good grasp of the big O notation with regards to assessing algorithm cost and running time?

I am doing this at college at the moment, but cant get my head around the whole aspect.

I know that the definition:

f(n) is O(g(n)) if there exists constants c and n0, such that
0 <= f(n) <= cg(n) for all n >= n0

I can see that for a certain value of n from n0 onwards this will hold true. But what does it tell me?

Also, is it true that Big Omega is "Best case" for an algorithm and Big Theta is "somewhere inbetween" Best Case(Omega) and Worst Case (Big O)?

Any insight you can pass on would be much appreciated.

Many thanks.

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cyberbob

f is in the "order of" g .

its *sorta* like saying a certain distance is in the order of inches or miles or lightyears... cept you're talking about the order of a function of n

baby example .

say you have a grid n wide and n tall.
define f(n) as the time taken to populate the grid
define g(n) = n * n

hence in this case f(n) = c*g(n) , c being the time taken to populate each cell.
ie the trivial case of the O function. with n0 = 1

in studying algorithms you take n as the length or size of the problem and g(n) being the complexity of the problem, eg populating your grid plus colouring in the bottom row say would be O ( c1 * n*n + c2 * n ) or O(n*n) as this action dominates (assuming c1>= c2)

dunno if this helps...

lisbon_lions

Thanks Bob,

So what we are saying is that we can forget about constants and trivial aspects from a function as this is minor in the overall context?

lets say c1 +2n + n*n would just be O(n*n) as this is dominant as n increases to bigger values?

Like for instance, lets value n at 5. 90+n + n*n, the dominant factor(n*n) would kick in from n=10 onwards. so in this example n0 would be 10 and from that point onward it would always hold true?

cyberbob

in my experience of studying complexity ( which is a while back now ) , you're never really concerned with relatively small n , so for brevity you'd talk about your example being O( n squared ) .

I suppose if you take away that relaxation , what your saying sounds right... would never have given it that much thought.