Analyzing algorithms based on their time and space requirements as input size grows.
Time and space complexity are the two primary measures we use to analyze the efficiency of an algorithm. They provide a standardized way to compare different approaches to solving a problem, independent of the hardware, programming language, or other environmental factors. Time complexity quantifies the amount of time an algorithm takes to run as a function of the length of the input. It's not about measuring the exact runtime in seconds, but rather about counting the number of basic operations performed. For instance, an algorithm with a time complexity of O(n) means the number of operations grows linearly with the input size 'n'. Similarly, O(n^2) means the operations grow quadratically, which is much slower for large 'n'. Space complexity refers to the total amount of memory space required by an algorithm to solve a problem, again as a function of the input size. This includes both the space needed for the input data and any auxiliary space used by the algorithm during its execution (like extra variables or data structures). Understanding these complexities is vital for writing scalable software. An algorithm that is fast on small inputs might become unacceptably slow or memory-intensive as the data grows. By analyzing complexity, we can predict this behavior and choose algorithms that remain performant and resource-efficient at scale.