University:
University of California San Diego
Course:
CSE 100/101 | Data Structures and Algorithms
Academic year:

2023
Views:

24

Pages:

2
Author:

anon

Understanding the Total Cost of a Sequence of Operations Using Amortized Analysis A technique for calculating the overall cost of a series of actions is amortized analysis. It is a technique for considering the average cost of each operation rather than just the highest possible cost. Instead of only concentrating on the worst case scenario, this gives us a more nuanced perspective of the overall cost of a set of activities. This article will define amortized analysis and examine one specific approach of performing it. We'll also examine the aggregate way of amortized analysis in more detail, using a dynamic array as an illustration. Amortized Cost: What Is It? The cost of a series of operations divided by the quantity of operations is the amortized cost. It is comparable to the amortized cost concept in finance, where you set aside money each month to pay for a significant expense like a car. You can understand the expense more accurately if it is broken up into smaller, easier to handle payments. If a car costs $6,000, for instance, you could save $100 every month for five years to pay for it. The car costs $100 a month to amortize, whereas the worst-case monthly cost is $6,000. Instead of only considering the worst case scenario, the amortized cost gives us a more complex picture of the cost of a significant expense. The Aggregate Amortized Analysis Method According to the aggregate method of amortized analysis, we can use the amortized cost definition to directly calculate the amortized cost. With this approach, we examine the price of each operation in turn before calculating the overall price. The amortized cost of a single operation is then calculated by dividing the total cost by the number of operations. Dynamic Array Illustration Let's examine the amortized analysis aggregate approach using a dynamic array as an illustration. Starting with a blank array, we'll make n calls to PushBack. We'll calculate the amortized cost of a single PushBack call. Although a call to PushBack has an O(n) worst-case time, let's define c[i] as the cost of the ith insertion. The costs from c[1] through c[n] are what we're interested in. Since we must write the i[i]th] element we are adding to the array, c[i] is obviously 1. If our capacity is exhausted, which occurs if the size is equal to the capacity, we must resize the array. If the previous insertion completely filled it, making it a full power of two, then this occurs. Let's assume that the array has a capacity of 8, and that 8 calls to PushBack are made, filling the array. Resizing costs O(n), yet a single call to PushBack costs O after amortization (1). This is due to the fact that the cost of resizing the array is divided by the quantity of PushBack calls, resulting in a lower cost per call. A useful method for determining the overall expense of a series of procedures is amortized analysis. This gives us a more nuanced perspective of the entire cost since it enables us to look at the average cost of each operation rather than just the worst case cost.