48 lines
2.6 KiB
Markdown
48 lines
2.6 KiB
Markdown
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---
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title: Big Theta Notation
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---
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## Big Theta Notation
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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.
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Take, for example, a function that searches an array for the value 0:
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```python
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def containsZero(arr): #assume normal array of length n with no edge cases
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for num x in arr:
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if x == 0:
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return true
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return false
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```
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1. 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)
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2. What's the worst case? If the array doesn't contain 0, we will have iterated through the whole array: O(n)
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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.
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Let's change our code a bit.
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```python
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def printNums(arr): #assume normal array of length n with no edge cases
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for num x in arr:
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print(x)
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```
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Can you think of a best case and worst case??
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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).
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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!
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If you're familiar with linked lists, test yourself and think about the runtimes for each of these functions!
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1. get
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2. remove
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3. add
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Things get even more interesting when you consider a doubly linked list!
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#### More Information:
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<!-- Please add any articles you think might be helpful to read before writing the article -->
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https://www.khanacademy.org/computing/computer-science/algorithms/asymptotic-notation/a/big-big-theta-notation
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https://stackoverflow.com/questions/10376740/what-exactly-does-big-%D3%A8-notation-represent
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https://www.geeksforgeeks.org/analysis-of-algorithms-set-3asymptotic-notations/
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