Big O notation is a simplified analysis of an algorithm’s efficiency. Big O notation gives us an algorithm’s complexity in terms of input size, N. It gives us a way to abstract the efficiency of our algorithm or code from the machines/computers they run on. We don’t care how powerful our machine is, but rather, the basic steps of the code. We can use big O to analyze both time and space. I will go over how we can use Big O to measure time complexity using Ruby for examples.
Types of measurement
There are a couple of ways to look at an algorithm’s efficiency. We can examine worst-case, best-case, and average-case. When we examine big O notation, we typically look at the worst-case. This isn’t to say the other cases aren’t as important.
General rules
- Ignore constants
5n ->O(n)
Big O notation ignores constants. For example, if you have a function that has a running time of 5n, we say that this function runs on the order of the big O of N. This is because as N gets large, the 5 no longer matters.
2. In the same way that N grows, certain terms “dominate” others
Here’s a list:
O(1) < O(logn) < O(n) < O(nlogn) < O(n²) < O(2^n) < O(n!)
We ignore low-order terms when they are dominated by high order terms.
Constant Time: O(1)
This basic statement computes x and does not depend on the input size in any way. This is independent of input size, N. We say this is a “Big O of one” or constant time.
total time = O(1) + O(1) + O(1) = O(1)
What happens when we have a sequence of statements? Notice that all of these are constant time. How do we compute big O for this block of code? We simply add each of their times and we get 3 * O(1). But remember we drop constants, so this is still big O of one.
Linear Time: O(n)
The run time of this method is linear or O(n). We have a loop inside of this method which iterates over the array and outputs the element. The number of operations this loop performs will change depending on the size of the array. For example, an array of size 6 will only take 6 iterations, while an array of 18 elements would take 3 times as long. As the input size increases, so will the runtime.Big O notation is a simplified analysis of an algorithm’s efficiency. Big O notation gives us an algorithm’s complexity in terms of input size, N. It gives us a way to abstract the efficiency of our algorithm or code from the machines/computers they run on. We don’t care how powerful our machine is, but rather, the basic steps of the code. We can use big O to analyze both time and space. I will go over how we can use Big O to measure time complexity using Ruby for examples.
Types of measurement
There are a couple of ways to look at an algorithm’s efficiency. We can examine worst-case, best-case, and average-case. When we examine big O notation, we typically look at the worst-case. This isn’t to say the other cases aren’t as important.
General rules
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