What Is Amortized Analysis Data Structure?


Heather Bennett

What Is Amortized Analysis Data Structure?

Amortized analysis is a technique used to analyze the performance of algorithms or data structures over a sequence of operations. It provides a way to average out the time complexity of individual operations, giving us an overall understanding of the efficiency of the algorithm or data structure.

Understanding Amortized Analysis

Amortized analysis is particularly useful when dealing with data structures that have operations with varying time complexities. It allows us to determine the average performance of these operations over a long period, even if some individual operations may be more costly than others.

This technique takes into account the fact that some expensive operations are followed by a series of cheaper ones, which balance out the overall cost. By considering this sequence of operations as a whole, we can determine an amortized time complexity that provides a more accurate representation of the actual cost.

The Aggregate Method

One common approach to perform amortized analysis is using the aggregate method. This method involves determining an upper bound on the total cost of a sequence of operations and dividing it by the number of operations to obtain an average cost.

To illustrate this concept, let’s consider an example with a dynamic array implementation. In this data structure, each time we run out of space and need to resize the array, it typically takes O(n) time where n is the current size. However, resizing happens infrequently compared to other cheaper array operations like insertion and deletion.

Assuming we allocate enough extra space during each resize to minimize future resizes, we can conclude that for every k insertions or deletions, there will be at most one resize operation. This means that even though resizing is an expensive operation, its cost can be spread across multiple insertions and deletions.

The Accounting Method

Another approach to amortized analysis is the accounting method. This method involves assigning credits or charges to each operation, such that the total credits assigned is always greater than or equal to the total charges.

By distributing the cost of expensive operations evenly across multiple cheaper operations, we can ensure that the overall cost is amortized. This allows us to perform costly operations without affecting the efficiency of subsequent operations, resulting in a more balanced performance.

Benefits of Amortized Analysis

Amortized analysis offers several benefits:

  • Accurate Performance Estimation: By considering the average cost over a sequence of operations, we get a more accurate estimation of the overall performance.
  • Better Understanding: It helps us understand how costs are distributed across different operations and how they affect each other.
  • Optimization Opportunities: By identifying expensive operations that can be amortized, we can optimize algorithms or data structures to improve their overall performance.


Amortized analysis is a powerful technique for analyzing algorithms or data structures with varying time complexities. It allows us to determine an average cost by considering a sequence of operations as a whole. Through methods like aggregate analysis and accounting methods, we can achieve a more accurate understanding of performance and optimize our designs accordingly.

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