Big O Notation Cheat Sheet

Understanding the performance of algorithms is crucial for both computer scientists and software engineers. One of the most effective tools for this is Big O notation. Let's delve into this subject, which is not just academic but also highly practical for anyone writing code.

Introduction to Big O Notation

According to mathematical definitions, Big O notation describes the upper bound of an algorithm's running time in the worst-case scenario. It's a part of a family of notations known as Bachmann–Landau notation or asymptotic notation. In layman's terms, Big O notation gives you a high-level understanding of the time complexity of an algorithm.

A Deep Dive into Various Complexity Classes

To gain a better understanding of how algorithms perform, let's explore the common complexity classes.

Constant Time: (O(1))

• Description: The running time remains constant regardless of the data set size.
• Efficiency: Highly efficient for any data set.
• Example Use Case: Accessing an array element by index.

Logarithmic Time: (O(\log N))

• Description: The data set size is halved with each iteration.
• Efficiency: Very efficient for large data sets.
• Example Use Case: Binary search algorithms.

Linear Time: (O(N))

• Description: The running time increases linearly with the size of the data set.
• Efficiency: Efficiency degrades as the data set grows.
• Example Use Case: Looping through a single-dimensional array.

Linearithmic Time: (O(N \log N))

• Description: The algorithm divides the data set and employs concurrency on independent lists.
• Efficiency: Efficient for medium to large-sized data sets.
• Example Use Case: Quick sort algorithm.

Polynomial Time: (O(N^2), O(N^3), \ldots)

• Description: Running time is proportional to some power of the size of the data set.
• Efficiency: Efficiency suffers significantly with larger data sets.
• Example Use Case: Nested loops, Bubble sort.

Exponential Time: (O(2^N))

• Description: Running time doubles with each additional element in the data set.
• Efficiency: Highly inefficient, especially for large data sets.
• Example Use Case: Recursive Fibonacci calculations.

Decoding the Symbols in Big O Notation

You may have noticed various symbols when discussing Big O. Here's what they mean:

Notation Glossary

• (O()): Describes the upper bound of the algorithm's complexity.
• (\Omega()): Represents the lower bound of the algorithm's complexity.
• (\Theta()): Specifies the exact bound of the algorithm's complexity.