Big O notation

In computer science, Big O notation is used to classify algorithms according to how their running time or space requirements grow as the input size grows.

Formally, let and be two functions defined on some subset of the real numbers. We say that is (or is of order ) if and only if there exists a positive constant such that for all sufficiently large values of , the absolute value of is at most multiplied by the absolute value of . That is,

for all x greater than some value x0.

The function is because as n grows larger, the term becomes dominant. The constants and lower-degree terms are irrelevant for large inputs.

Here's how you might express this in Python code:

def f(n):
    return 6*n**2 + 2*n + 5

print(f(10)) # Output: 625
print(f(100)) # Output: 60205
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As you can see, as n increases, the output of the function increases much faster than linearly – it increases quadratically. This function would be considered "worse" than a function that increases linearly (i.e., ) but "better" than one that increases exponentially (i.e., ).

Big O notation is used in many fields, but its most common application is in computer science for analyzing algorithms. It provides an upper bound on the time complexity of an algorithm, which helps programmers and developers understand how their code will scale with increasing input size.

For example, when sorting a list of items, a sorting algorithm with a time complexity of () (like Bubble Sort) will perform significantly worse than an algorithm with a time complexity of O() (like Merge Sort) as the size of the list increases.

In addition to time complexity, Big O notation can also be used to describe space ccomplexity – howthe memory usage of an algorithm grows with the size of the input. This can be crucial when working with large data sets or in environments where memory is limited.

Understanding Big O notation and being able to analyze algorithms using it is a fundamental skill for any software engineer or developer. It allows them to make informed decisions about which algorithms to use in different situations, and can help them optimize their code for performance and efficiency.