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Re: BigO Notation  What is it good for?by blokhead (Monsignor) 
on Sep 15, 2006 at 17:54 UTC ( #573208=note: print w/ replies, xml )  Need Help?? 
... it [bigO] describes how the algorithm scales (performs) in the worst case scenario as it is is run with more input ...There is a common misconception that BigO means worstcase, BigOmega means bestcase, BigTheta means averagecase, etc. Well, kinda. In many common cases, that is often what is meant. But this can be an imprecise mental model. Let me propose a different interpretation. For now let's ignore the distinction between bestcase, worstcase, averagecase running times. So imagine an algorithm that takes exactly the same amount of time on all inputs of the same size (bestcase, worstcase and averagecase all the same). Then:
So why is BigO commonly associated with worstcase running times, and why is that imprecise? It's because when considering the worst possible case, it is natural to give a limit on how bad that worst case can be, not how good it can me. That is, we want to give an upper bound on its degree of badness. Similarly, we often want to give a lower bound on how good the bestcase is (i.e, even on good inputs, there's still a limit on how fast the algorithm can go; what is that limit?), so BigOmega gets associated with bestcase. That's why BigO gets associated with worstcase running times, and BigOmega with bestcase. And it's true that if someone just says "the running time" is O(n^2), then n^2 is indeed "closer" to the worstcase running time than to the bestcase running time, in the sense that n^2 is "bigger" than all possible running times, and the worstcase running time is "bigger" than the bestcase running time. But O(n^2) doesn't mean that the worstcase running time actually is n^2, just that it is at most n^2. And it's not wrong or illegal for me to express (for instance) a lower bound on the worstcase running time (i.e, "the worstcase is at least this bad"). In fact, this kind of bound can be quite useful. BigO and friends are just expressing bounds/limits on a function. I can give both upper bounds and lower bounds for any function, whether that function expresses the worstcase, bestcase, or averagecase running time of an algorithm. My second point, which I already hinted at, deals with how close the upper/lower bounds are to the actual running time. Remember, BigO expresses an upper limit over your running time, but it doesn't need to be close to the running time! Here's an example. Again, for simplicity, let's just assume that averagecase = bestcase = worstcase running times. I have two algorithms. My local CS major tells me that algorithm A's runtime is O(n^3), but algorithm B's runtime is O(n^2). Which is better? Well, all I know is that algorithm A takes no more than n^3 steps, and algorithm B takes no more than n^2 steps. It may be the case that algorithm A takes much fewer than n^3 steps. Maybe it was a difficult algorithm to analyze, and it actually takes exactly n steps, but our CS major could only prove that it took less than n^3. The upper bound of O(n^3) is still absolutely correct. And it may be the case that algorithm B takes exactly n^2 steps on its inputs. It was an easy algorithm to analyze and the CS major could exactly count the number of steps. The upper bound here of O(n^2) is still correct. In this example, algorithm B has the more "attractive" bigO, but algorithm A is always faster. (Update: fixed this sentence according to jpearl's comment) So while BigO gives you an upper bound, it does not tell you how good that bound is. It may be a very very rough/sloppy overestimate. So it's nice to be able to have corresponding lower bounds as well. If the upper and lower bounds for some running time are close, you know that the actual running time must live in the small space between these two bounds. This is why I mentioned that asking for a lower bound on the worstcase running time could be a useful thing. This is why we get things like BigOmega and BigTheta, but for that you'll have to look elsewhere. This reply is already way too long ;) blokhead
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