Big O Notation?

For beginner and those considering investing some time into the study of computer science, this question serves as the start of a different way to look at solving problems. Maybe the term is familiar; if it isn't, let's fix that.

Complexity Analysis

Complexity analysis is a way to measure the efficiency of an algorithm. It describes the relationship between the size of the input and the number of operations required to solve a problem. In other words, it tells us how the running time of an algorithm grows as the size of the input increases.

Big O Notation

The most common way to express the complexity of an algorithm is using Big O notation. Big O notation describes the upper bound on the number of operations required by an algorithm. It tells us the worst-case scenario for the number of operations required to solve a problem, regardless of the actual input.

For example, let's say we have an algorithm that takes an array of integers and finds the largest number in the array. The number of operations required to solve this problem will be directly proportional to the size of the array. If the array has 10 elements, the algorithm will take 10 operations. If the array has 100 elements, the algorithm will take 100 operations. In Big O notation, we would say that the complexity of this algorithm is O(n), where n is the size of the array.

There are several common complexities that you will come across when analyzing algorithms. Here are a few:

Constant time: O(1)

The number of operations required to solve the problem does not change as the size of the input increases.

Logarithmic time: O(log n)

The number of operations required to solve the problem increases logarithmically with the size of the input.

Linear time: O(n)

The number of operations required to solve the problem is directly proportional to the size of the input.

Log-linear time: O(n log n)

The number of operations required to solve the problem is a logarithmic function of the size of the input.

Quadratic time: O(n^2)

The number of operations required to solve the problem is a square function of the size of the input.

When analyzing algorithms, it's important to keep in mind that Big O notation describes the worst-case scenario. There may be cases where the actual number of operations required is much less than the upper bound described by Big O notation.

In summation, Big O notation is a way to measure the efficiency of an algorithm by describing the relationship between the size of the input and the number of operations required to solve a problem. It helps us to understand how the running time of an algorithm grows as the size of the input increases. By understanding the complexities of different algorithms, we can make informed decisions about which algorithm to use for a given problem.