What is the difference between big O, big omega, and big theta notation?

1 Big O notation

Big O notation is the most common way to express the upper bound of an algorithm's running time. It tells you how fast your algorithm grows as the input size increases. For example, if your algorithm has a running time of O(n^2), it means that it takes at most n^2 steps to complete, where n is the input size. Big O notation helps you compare different algorithms and choose the one that performs better in the worst case scenario.

2 Big omega notation

Big omega notation is the opposite of big O notation. It tells you the lower bound of an algorithm's running time. It means that your algorithm takes at least a certain number of steps to complete, regardless of the input size. For example, if your algorithm has a running time of omega(n), it means that it takes at least n steps to finish, where n is the input size. Big omega notation helps you measure the best case performance of your algorithm and identify the minimum resources required.

3 Big theta notation

Big theta notation is a combination of big O and big omega notation. It tells you the tight bound of an algorithm's running time. It means that your algorithm takes a constant factor of steps to complete, within a range of input sizes. For example, if your algorithm has a running time of theta(n log n), it means that it takes between c1 * n log n and c2 * n log n steps to finish, where c1 and c2 are constants and n is the input size. Big theta notation helps you describe the average case performance of your algorithm and the exact growth rate.

4 How to use them

To use these notations, you need to identify the dominant term in your algorithm's running time. The dominant term is the one that has the highest degree or the highest growth rate. For example, if your algorithm has a running time of 3n^2 + 5n + 2, the dominant term is 3n^2. You can ignore the other terms and the coefficients, and write your algorithm's running time as O(n^2), omega(n^2), or theta(n^2), depending on the context.

5 Examples of common notations

Common notations and their meanings can be illustrated with the following examples. O(1) implies a constant number of steps, regardless of the input size; for instance, accessing an element in an array by its index. O(log n) stands for a logarithmic number of steps, meaning that it halves the input size at each step; for example, binary search in a sorted array. O(n) means a linear number of steps, proportional to the input size; for instance, traversing an array or a linked list. O(n log n) indicates a linearithmic number of steps, combining a linear and a logarithmic operation; such as merge sort or quick sort. O(n^2) denotes a quadratic number of steps, performing a linear operation for each element in the input; such as bubble sort or insertion sort. Finally, O(2^n) defines an exponential number of steps, doubling the number of steps for each additional element in the input; for instance, the naive recursive solution for the Fibonacci sequence.

6 Why they matter

Understanding these notations is important for software design, because they help you evaluate the trade-offs between different solutions and optimize your code for speed and space. They also help you communicate your ideas and expectations with other developers and stakeholders. By using these notations, you can express the complexity and efficiency of your algorithms in a concise and standardized way.

7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?