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# Algorithmic Complexity ## Definition **Algorithmic complexity** is a measure of the amount of resources (time, space) required by an algorithm to solve a problem of a given size. * **Time complexity:** the amount of time required by an algorithm to complete. * **Space complexity:** the amou...

# Algorithmic Complexity ## Definition **Algorithmic complexity** is a measure of the amount of resources (time, space) required by an algorithm to solve a problem of a given size. * **Time complexity:** the amount of time required by an algorithm to complete. * **Space complexity:** the amount of memory space required by an algorithm to complete. Complexity is typically 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. ## Common complexities Here are some common Big O complexities, ordered from fastest to slowest: | Complexity | Name | Description | Example | | :---------------- | :------------ | :------------------------------------------------------------------------------------------------------------------------------------------------- | :--------------------------------------------------------------------- | | $O(1)$ | Constant | The algorithm takes the same amount of time or space regardless of the input size. | Accessing an element in an array by index. | | $O(\log n)$ | Logarithmic | The algorithm's resource usage increases logarithmically with the input size. | Binary search in a sorted array. | | $O(n)$ | Linear | The algorithm's resource usage increase linearly with the input size. | Searching for an element in an unsorted array. | | $O(n \log n)$ | Linearithmic | The algorithm's resource usage increases linearly with the input size, multiplied by a logarithmic factor. | Merge sort, quicksort (average case). | | $O(n^2)$ | Quadratic | The algorithm's resource usage increases quadratically with the input size. | Bubble sort, insertion sort. | | $O(n^3)$ | Cubic | The algorithm's resource usage increases cubically with the input size. | Matrix multiplication. | | $O(2^n)$ | Exponential | The algorithm's resource usage doubles with each addition to the input size. | Finding all subsets of a set. | | $O(n!)$ | Factorial | The algorithm's resource usage grows factorially with the input size. | Finding all permutations of a set. | ## Importance Understanding algorithmic complexity is crucial for: * **Choosing the right algorithm:** Selecting an algorithm with lower complexity can significantly improve performance, especially for large input sizes. * **Predicting performance:** Estimating the resource usage of an algorithm helps predict its performance and identify potential bottlenecks. * **Optimizing code:** Analyzing the complexity of code can reveal areas where optimizations can have the greatest impact. ## Tips Here are some tips for analyzing and improving algorithmic complexity: * **Identify bottlenecks:** Pinpoint the parts of the algorithm that contribute the most to its complexity. * **Consider data structures:** Choosing the right data structure can significantly impact the complexity of an algorithm. * **Apply algorithmic techniques:** Techniques like divide and conquer, dynamic programming, and memoization can often reduce complexity. * **Optimize code:** Simple code optimizations like loop unrolling and caching can improve performance, but they typically don't change the Big O complexity.