Big O notation is a mathematical notation used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity. In computer science, it is primarily used to classify algorithms according to how their run time or space requirements grow as the input size grows. This notation provides a high-level understanding of the algorithm's efficiency, particularly in the worst-case scenario.
Key Points of Big O Notation:
Purpose:
Time Complexity: Describes how the runtime of an algorithm increases with the size of the input.
Space Complexity: Describes how the memory usage of an algorithm increases with the size of the input.
Notation:
Big O notation is denoted as $$O(f(n))$$, where $$f(n)$$ is a function that represents the number of operations an algorithm performs to solve a problem of size $$n$$.
Upper Bound:
Big O notation provides an upper bound on the growth rate of the function, meaning it describes the worst-case scenario for the algorithm's performance.
Common Big O Notations:
O(1): Constant time complexity, where the algorithm's runtime is constant regardless of the input size (e.g., accessing an element in an array by index).
O(n): Linear time complexity, where the algorithm's runtime grows linearly with the input size (e.g., linear search through an array).
O(log n): Logarithmic time complexity, where the algorithm's runtime grows logarithmically with the input size (e.g., binary search in a sorted array).
O(n^2): Quadratic time complexity, where the algorithm's runtime grows quadratically with the input size (e.g., bubble sort).
O(2^n): Exponential time complexity, where the algorithm's runtime doubles with each addition to the input data set (e.g., recursive calculation of Fibonacci numbers).
Importance:
Algorithm Comparison: Allows for the comparison of the efficiency of different algorithms.
