amortized analysis

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In
computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of , , and . Computer science ...
, amortized analysis is a method for
analyzing Analysis is the process of breaking a complexity, complex topic or Substance theory, substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Ari ...
a given algorithm's
complexity Complexity characterises the behaviour of a system A system is a group of interacting Interaction is a kind of action that occurs as two or more objects have an effect upon one another. The idea of a two-way effect is essential in the con ...
, 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 ''per operation'', rather than ''per algorithm'', can be too pessimistic. While certain operations for a given algorithm may have a significant cost in resources, other operations may not be as costly. The amortized analysis considers both the costly and less costly operations together over the whole series of operations of the algorithm. 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 Robert Endre Tarjan (born April 30, 1948) is an American computer scientist A computer scientist is a person A person (plural people or persons) is a being that has certain capacities or attributes such as reason, morality, consciousness or s ...
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 tree In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of , ...

s and union operations. However, it is now ubiquitous and comes into play when analyzing many other algorithms as well.

# Method

Amortized analysis requires knowledge of which series of operations are possible. This is most commonly the case with
data structure In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of ...

s, which have
state State may refer to: Arts, entertainment, and media Literature * ''State Magazine'', a monthly magazine published by the U.S. Department of State * The State (newspaper), ''The State'' (newspaper), a daily newspaper in Columbia, South Carolina, Un ...
that persists between operations. The basic idea is that a worst-case operation can alter the state in such a way that the worst case cannot occur again for a long time, thus "amortizing" its cost. There are generally three methods for performing amortized analysis: the aggregate method, the accounting method, and the potential method. All of these give correct answers; the choice of which to use depends on which is most convenient for a particular situation. *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''. *The accounting method is a form of aggregate analysis which assigns to each operation an ''amortized cost'' which may differ from its actual cost. Early operations have an amortized cost higher than their actual cost, which accumulates a saved "credit" that pays for later operations having an amortized cost lower than their actual cost. Because the credit begins at zero, the actual cost of a sequence of operations equals the amortized cost minus the accumulated credit. Because the credit is required to be non-negative, the amortized cost is an upper bound on the actual cost. Usually, many short-running operations accumulate such credit in small increments, while rare long-running operations decrease it drastically. *The potential method is a form of the accounting method where the saved credit is computed as a function (the "potential") of the state of the data structure. The amortized cost is the immediate cost plus the change in potential.

# Examples

## Dynamic array

Consider a
dynamic array In computer science, a dynamic array, growable array, resizable array, dynamic table, mutable array, or array list is a random access, variable-size list data structure that allows elements to be added or removed. It is supplied with standard libra ...

that grows in size as more elements are added to it, such as in Java or in C++. If we started out with a dynamic array of size 4, we could push 4 elements onto it, and each operation would take
constant time In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of Algor ...
. Yet pushing a fifth element onto that array would take longer as the array would have to create a new array of double the current size (8), copy the old elements onto the new array, and then add the new element. The next three push operations would similarly take constant time, and then the subsequent addition would require another slow doubling of the array size. In general if we consider an arbitrary number of pushes ''n'' + 1 to an array of size ''n'', we notice that push operations take constant time except for the last one which takes time to perform the size doubling operation. Since there were ''n'' + 1 operations total we can take the average of this and find that pushing elements onto the dynamic array takes: $\tfrac = \Theta\left(1\right)$, constant time.

## Queue

Shown is a Ruby implementation of a
Queue__NOTOC__ Queue () may refer to: * Queue area, or queue, a line or area where people wait for goods or services Arts, entertainment, and media *''ACM Queue'', a computer magazine * The Queue (publishing company), an American publishing company * ...
, a FIFO data structure: class Queue def initialize @input = [] @output = [] end def enqueue(element) @input << element end def dequeue if @output.empty? while @input.any? @output << @input.pop end end @output.pop end end The enqueue operation just pushes an element onto the input array; this operation does not depend on the lengths of either input or output and therefore runs in constant time. However the dequeue operation is more complicated. If the output array already has some elements in it, then dequeue runs in constant time; otherwise, dequeue takes time to add all the elements onto the output array from the input array, where ''n'' is the current length of the input array. After copying ''n'' elements from input, we can perform ''n'' dequeue operations, each taking constant time, before the output array is empty again. Thus, we can perform a sequence of ''n'' dequeue operations in only time, which implies that the amortized time of each dequeue operation is . 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 algorithm In computer science Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application. Computer science is the study of Algori ...
s commonly use amortized analysis.

# References

* {{Use dmy dates, date=June 2019 Analysis of algorithms Articles with example Ruby code