In computer science, amortized analysis is a method for analyzing a given algorithm's complexity, or how much of a resource, especially time or memory, it takes to execute. The motivation for amortized analysis is that looking at the worst-case run time can be too pessimistic. Instead, amortized analysis averages the running times of operations in a sequence over that sequence.
For a given operation of an algorithm, certain situations (e.g., input parametrizations or data structure contents) may imply a significant cost in resources, whereas other situations may not be as costly. The amortized analysis considers both the costly and less costly operations together over the whole sequence of operations. This may include accounting for different types of input, length of the input, and other factors that affect its performance.
History
Amortized analysis initially emerged from a method called aggregate analysis, which is now subsumed by amortized analysis. The technique was first formally introduced by Robert Tarjan in his 1985 paper Amortized Computational Complexity, which addressed the need for a more useful form of analysis than the common probabilistic methods used. Amortization was initially used for very specific types of algorithms, particularly those involving binary trees and union operations. However, it is now ubiquitous and comes into play when analyzing many other algorithms as well.
- Aggregate analysis determines the upper bound T(n) on the total cost of a sequence of n operations, then calculates the amortized cost to be T(n) / n.
Alternatively, we can charge the cost of copying any item from the input array to the output array to the earlier enqueue operation for that item. This charging scheme doubles the amortized time for enqueue but reduces the amortized time for dequeue to .
Common use
- In common usage, an "amortized algorithm" is one that an amortized analysis has shown to perform well.
- Online algorithms commonly use amortized analysis.