Amortized Time Complexity: Definition and Examples

Amortized Time Complexity: Definition and Examples

Amortized time complexity measures the average cost per operation over a sequence of operations on a data structure, even when some individual operations are expensive. Unlike worst-case analysis (which looks at a single operation in isolation) or average-case analysis (which relies on probability), amortized analysis guarantees that any sequence of n operations will take at most a certain total time, regardless of the order or pattern of operations. Key Idea Some operations are cheap most of the time but occasionally trigger an expensive "cleanup" or "resize" step. Amortized analysis spreads the cost of these rare expensive operations across the many cheap ones that precede them, showing that the average cost per operation remains low. Formally, if a sequence of m operations costs at most T(m) total time, the amortized cost per operation is T(m) / m. Three Methods of Amortized Analysis There are three standard techniques for performing amortized analysis: Aggregate Met…