# What Is Amortized Cost in Data Structure?

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Larry Thompson

What Is Amortized Cost in Data Structure?

In data structure, amortized cost is a method used to analyze the performance of an algorithm or data structure over a sequence of operations. It provides a way to average out the cost of individual operations and gives us a more accurate understanding of the overall efficiency.

When analyzing algorithms, it is not enough to simply consider their worst-case or average-case performance. In many cases, the costs associated with individual operations can vary widely. For example, some operations may be very fast while others may be slower.

## Understanding Amortization

To better understand amortized cost, let’s consider an example using a dynamic array. A dynamic array is an array with a variable size that can grow or shrink as needed.

When we insert elements into a dynamic array, we usually allocate more memory than necessary to accommodate future insertions. This extra memory allows us to avoid frequent reallocations and copying of elements when the array reaches its capacity.

The cost of inserting an element into a dynamic array can vary depending on whether we need to allocate new memory or if there is already enough space available. In some cases, inserting an element might take constant time (O(1)), while in others it might take linear time (O(n)) due to resizing and copying elements.

## Amortized Analysis

Amortized analysis allows us to determine the average cost per operation over a sequence of operations. It takes into account both the cheap and expensive operations and provides us with a more accurate understanding of the overall efficiency.

There are different techniques for performing amortized analysis, such as aggregate analysis and accounting methods. These methods assign some “credit” or “debt” to each operation and distribute it among other operations to ensure that the total cost is evenly distributed.

## Types of Amortized Costs

There are three common types of amortized costs:

• Constant Amortized Cost: This means that each operation has a constant cost, regardless of the input size or previous operations. Examples include adding an element to an ArrayList or pushing an element onto a stack.
• Logarithmic Amortized Cost: This means that the cost of an operation grows logarithmically with the input size.

Examples include inserting elements into a binary search tree or performing operations on a skip list.

• Linear Amortized Cost: This means that the cost of an operation grows linearly with the input size. Examples include inserting elements into a dynamic array or resizing a hash table.

## Benefits of Amortized Analysis

Amortized analysis allows us to make more informed decisions when choosing data structures or algorithms. By considering the amortized cost, we can avoid potential performance pitfalls and choose solutions that provide consistent and efficient performance over time.

In conclusion, amortized cost provides us with a way to analyze the overall efficiency of algorithms and data structures by considering their performance over a sequence of operations. It allows us to account for variations in individual operations and make better-informed decisions when designing applications.

If you found this article helpful, check out our other tutorials on various data structures and algorithms!