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Big O notation provides an upper bound to a function whereas Big Theta provides a tight bound. However I find that Big O notation is typically (and informally) taught and used when they really mean Big Theta.

e.g. "Quicksort is O(N^2)" can turned into the much stronger statement "Quicksort is Θ(N^2)"

While usage of Big O is technically correct, wouldn't a more prevalent use of Big Theta be more expressive and lead to less confusion? Is there some historical reason why this Big O is more commonly used?

Wikipedia notes:

Informally, especially in computer science, the Big O notation often is permitted to be somewhat abused to describe an asymptotic tight bound where using Big Theta Θ notation might be more factually appropriate in a given context.

tskuzzy
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2 Answers2

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Because you are usually just interested in the worst case when analyzing the performance. Thus, knowing the upper bound is sufficient.

When it runs faster than expected for a given input - that is ok, it's not the critical point. It's mostly negligible information.

Some algorithms, as @Peter Taylor noted, don't have a tight bound at all. See quicksort for example which is O(n^2) and Omega(n).

Moreover, tight bounds are often more difficult to compute.

See also:

Community
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Falcon
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9

One reason is that there are many cases in which Θ just isn't known. For instance, Matrix multiplication is O(n^2.376) but there's no known tight bound. Sure, as far as I can tell, there is a tight bound for Matrix multiplication, but we don't know its value.

MSalters
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