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# Algorithmic complexity ### Definition Algorithmic complexity is a measure of the amount of resources (typically time and space) required by an algorithm to solve a problem of a given size. It is usually expressed using Big O notation, which describes the upper bound of the growth rate of the alg...

# Algorithmic complexity ### Definition Algorithmic complexity is a measure of the amount of resources (typically time and space) required by an algorithm to solve a problem of a given size. It is usually expressed using Big O notation, which describes the upper bound of the growth rate of the algorithm's resource usage as the input size increases. ### Time complexity Time complexity refers to the amount of time an algorithm takes to run as a function of the input size. It is typically expressed using Big O notation, which provides an upper bound on the growth rate of the algorithm's execution time. **Common time complexities:** * **O(1)**: Constant time. The algorithm's execution time does not depend on the input size. * **O(log n)**: Logarithmic time. The algorithm's execution time increases logarithmically with the input size. * **O(n)**: Linear time. The algorithm's execution time increases linearly with the input size. * **O(n log n)**: Linearithmic time. The algorithm's execution time increases linearly with the input size multiplied by the logarithm of the input size. * **O(n^2)**: Quadratic time. The algorithm's execution time increases quadratically with the input size. * **O(2^n)**: Exponential time. The algorithm's execution time increases exponentially with the input size. * **O(n!)**: Factorial time. The algorithm's execution time increases factorially with the input size. ### Space complexity Space complexity refers to the amount of memory space an algorithm requires as a function of the input size. It is also typically expressed using Big O notation, which provides an upper bound on the growth rate of the algorithm's memory usage. **Factors affecting space complexity:** * Input data size * Auxiliary data structures * Temporary variables * Recursion depth ### Big O notation Big O notation is a mathematical notation used to describe the asymptotic behavior of functions. In the context of algorithmic complexity, it provides an upper bound on the growth rate of an algorithm's resource usage (time or space) as the input size increases. **Key concepts:** * **Upper bound**: Big O notation describes the worst-case scenario for an algorithm's resource usage. * **Asymptotic behavior**: Big O notation focuses on the growth rate of the algorithm's resource usage as the input size approaches infinity. * **Dominant terms**: Big O notation only considers the dominant terms in the expression for the algorithm's resource usage, ignoring constant factors and lower-order terms. **Common Big O notations:** * O(1): Constant * O(log n): Logarithmic * O(n): Linear * O(n log n): Linearithmic * O(n^2): Quadratic * O(2^n): Exponential * O(n!): Factorial ### Example Consider the following algorithm to find the maximum element in an array: ``` function findMax(array) { let max = array; for (let i = 1; i < array.length; i++) { if (array[i] > max) { max = array[i]; } } return max; } ``` **Time complexity analysis:** * The algorithm iterates through the array once, performing a constant amount of work for each element. * Therefore, the time complexity is O(n), where n is the size of the array. **Space complexity analysis:** * The algorithm uses a constant amount of extra space to store the `max` variable. * Therefore, the space complexity is O(1). ### Importance of algorithmic complexity Understanding algorithmic complexity is crucial for designing efficient algorithms and choosing the right algorithm for a particular problem. By analyzing the time and space complexity of different algorithms, developers can make informed decisions about which algorithm will perform best for a given input size. This is especially important when dealing with large datasets or performance-critical applications.