title: Big Theta Notation
Big Theta Notation
Big Omega tells us the lower bound of the runtime of a function, and Big O tells us the upper bound. Often times, they are different and we can’t put a guarantee on the runtime – it will vary between the two bounds and the inputs. But what happens when they’re the same? Then we can give a theta (Θ) bound – our function will run in that time, no matter what input we give it. In general, we always want to give a theta bound if possible because it is the most accurate and tightest bound. If we can’t give a theta bound, the next best thing is the tightest O bound possible.
Take, for example, a function that searches an array for the value 0:
def containsZero(arr): #assume normal array of length n with no edge cases for num x in arr: if x == 0: return true return false
- What’s the best case? Well, if the array we give it has 0 as the first value, it will take constant time: Ω (1)
- What’s the worst case? If the array doesn’t contain 0, we will have iterated through the whole array: O(n)
We’ve given it an omega and O bound, so what about theta? We can’t give it one! Depending on the array we give it, the runtime will be somewhere in between constant and linear.
Let’s change our code a bit.
def printNums(arr): #assume normal array of length n with no edge cases for num x in arr: print(x)
Can you think of a best case and worst case??
I can’t! No matter what array we give it, we have to iterate through every value in the array. So the function will take AT LEAST n time (Ω(n)), but we also know it won’t take any longer than n time (O(n)). What does this mean? Our function will take exactly n time: Θ(n).
If the bounds are confusing, think about it like this. We have 2 numbers, x and y. We are given that x <= y and that y <= x. If x is less than or equal to y, and y is less than or equal to x, then x has to equal y!
If you’re familiar with linked lists, test yourself and think about the runtimes for each of these functions!
Things get even more interesting when you consider a doubly linked list!