What is amortized analysis?

Amortized analysis gives the average cost per operation over a sequence of operations, even if some individual operations are expensive. Unlike average-case analysis (probabilistic), amortized analysis is a worst-case guarantee for the average cost over any sequence. Methods: (1) Aggregate method: compute the total cost of n operations, divide by n. Example: dynamic array — n pushes cost at most 3n total (each element is moved at most twice across all doubly-capacity resizings). Amortized cost = 3n/n = O(1) per push; (2) Accounting method: assign amortized costs to operations; some operations "pay extra" (credit) that covers future expensive operations. Each operation pays its actual cost + charges credit; (3) Potential method: define a potential function Φ(state); amortized cost of operation = actual cost + ΔΦ. Classic examples: (1) Dynamic array (ArrayList): push O(1) amortized despite occasional O(n) resize; (2) Splay tree: self-adjusting BST — O(log n) amortized; (3) Fibonacci heap: decrease-key O(1) amortized; (4) Binary counter: increment O(1) amortized despite occasional carry propagation; (5) Union-Find: nearly O(1) amortized with path compression + union by rank.