About insertion sort time complexity

Question

In my text book, it says that time complexity of insertion sort is Θ(n ² ). I couldn't understand that since insertion sort's best case is O(n). I know big theta is lower and upper bound both. So is it correct that time complexity of insertion sort is O(n ² ) not Θ(n ² )? Sorry to my poor english thx.

Answer 1

O vs. Θ isn't worst-case vs. all cases; it's the difference between <= and = .

For example the algorithm Merge sort is O(n^2), but it's not Θ(n^2) because it's faster than that.

Answer 2

You have to be careful when talking about "the" time complexity of a piece of code, because there isn't one single time complexity you can point at. A piece of code can have a best-case time complexity, a worst-case time complexity, an average-case time complexity, etc., and they don't have to all be the same.

Saying that "insertion sort is Θ(n ² )" is a bit sloppy because it's not insertion sort itself that's Θ(n ² ), but rather its worst-case runtime. Typically, if you say that a piece of code is O(f(n)), you're saying that its runtime is O(f(n)), and usually that implies that you're talking about the worst-case runtime.

It's true that the best-case runtime of insertion sort is Θ(n), which happens when the input is already sorted. The worst-case runtime is Θ(n ² ), which happens on a reverse-sorted list. Assuming the input is a random permutation of n elements, the average-case runtime is also Θ(n ² ). Those more precise statements are probably better to make than "insertion sort is O(n)" or "insertion sort is Θ(n ² )," since they capture more about the runtime of insertion sort.

Do note that just using O, Θ, or Ω doesn't automatically mean that you're talking about best/worst/average case complexity. You can use Θ notation to talk about a best-case, worst-case, or average-case runtime, for example.

Answer 3

When generally speaking of ' the time complexity of an algorithm ' one usually speaks of all types of inputs , not restricted to eg worse-case or best-case scenarios.

In this general case, for all types of inputs, one case (the worst-case) is enough to determine the complexity class to be the worse class .

So it does not help that the best-case here yields Θ(n) when we speak of the time complexity for all inputs .

However you can also analyze an algorithm explicitly for a specific scenario, eg only for the best-case or worst-case and so on. Then you restrict the algorithm and only allow such types of input.