Big O notation

Big-O-notation

comparison_computational_complexity

Big O notation

Big O notation is a mathematical notation used to describe the asymptotic behavior of functions. It is particularly useful in computer science for analyzing the time complexity and space complexity of algorithms. Big O notation provides an upper bound on the growth rate of a function, allowing for the comparison of the efficiency of different algorithms.

Definition

Big O notation is defined as follows: A function f(n) is said to be O(g(n)) if there exist positive constants c and n0 such that for all n ≥ n0, f(n) ≤ c * g(n). In this context, f(n) represents the function being analyzed, and g(n) is the comparison function.

Common Big O Notations

Several common Big O notations are frequently used to describe the performance of algorithms:

• O(1): Constant time complexity. The algorithm's performance is independent of the input size.
• O(log n): Logarithmic time complexity. The algorithm's performance grows logarithmically with the input size.
• O(n): Linear time complexity. The algorithm's performance grows linearly with the input size.
• O(n log n): Linearithmic time complexity. The algorithm's performance grows in proportion to n log n.
• O(n^2): Quadratic time complexity. The algorithm's performance grows quadratically with the input size.
• O(2^n): Exponential time complexity. The algorithm's performance grows exponentially with the input size.
• O(n!): Factorial time complexity. The algorithm's performance grows factorially with the input size.

Usage in Computer Science

Big O notation is widely used in algorithm analysis to classify algorithms according to their performance and resource usage. It helps in understanding the scalability of algorithms and in making informed decisions about which algorithm to use in a given context.

Related Concepts

• Little o notation
• Big Theta notation
• Big Omega notation
• Time complexity
• Space complexity
• Algorithm
• Asymptotic analysis

See Also

• Algorithm
• Computational complexity theory
• Data structure
• P versus NP problem
• Sorting algorithm