Question

a. The number of comparisons in the best case of a bubble sort algorithm, as given in this chapter, is \(O\left(n^{2}\right) .\) Show that the following version of the bubble sort algorithm reduces the number of comparisons in the best case of the bubble sort algorithm to \(O(n)\) //list – list to be sorted //elemType – type of the list elements //length – length of the list bool isSorted = false; for (int iteration = 1; (iteration < length) && !isSorted; iteration++) { isSorted = true; //assume that the sublist is sorted for (int index = 0; index < length - iteration; index++) { if (list[index] > list[index + 1]) { elemType temp = list[index]; list[index] = list[index + 1]; list[index + 1] = temp; isSorted = false; } } } b. Using the algorithm given in part (a), find the number of iterations that are needed to sort the following list: 65,14,52,43,75,25,80,90,95

Short answer

The modified bubble sort is best-case complexity is O(n). For the list, 5 iterations are needed to sort.

Step-by-step solution

Understand the Bubble Sort Modification

In this modified version of Bubble Sort, an 'isSorted' flag is introduced. The flag, initially set to true in each iteration, assumes the list is sorted. During the iteration over the list, if any swap is made, the flag is set to false, indicating that the sublist was not sorted and another iteration is required. If the flag remains true after a complete pass without swaps, the algorithm stops because the list is sorted.

Analyze Best Case Comparisons

In the best case scenario, the list is already sorted. Here, the first iteration over the list will result in no swaps, keeping 'isSorted' true. Recognizing this, the algorithm then stops, as further iterations are unnecessary. The result is a single pass through the list, making this best-case scenario linear.

Determine Best Case Complexity

Since the algorithm only requires a single pass through the list when already sorted, the number of comparisons in the best case is linear with respect to the number of elements in the list. Thus, the algorithm performs comparisons in the best case.

Apply Bubble Sort to the Given List

List: 65, 14, 52, 43, 75, 25, 80, 90, 95. Initially, 'isSorted' is false. You will execute iterations checking and performing swaps where needed until no more swaps occur, marking the list as sorted.

First Iteration

Traverse the list: 65 swaps with 14, 52 stays in place, 43 swaps with 52, 75 stays in place, 25 swaps with 75, 80, 90, 95 remain, resulting in the list: 14, 65, 43, 52, 25, 75, 80, 90, 95. Set 'isSorted' to false, as swaps occurred.

Second Iteration

Traverse: 65 swaps with 43, 52 stays in place, 52 swaps with 25, resulting list: 14, 43, 65, 25, 52, 75, 80, 90, 95. 'isSorted' remains false as swaps occurred.

Third Iteration

Traverse: 65 swaps with 25, resulting list: 14, 43, 25, 52, 65, 75, 80, 90, 95. 'isSorted' remains false as swaps occurred.

Fourth Iteration

Traverse: 43 swaps with 25, list is 14, 25, 43, 52, 65, 75, 80, 90, 95. 'isSorted' remains false.

Fifth Iteration

No swaps needed, list remains 14, 25, 43, 52, 65, 75, 80, 90, 95. Set 'isSorted' to true and stop the sort.

Key concepts

Sorting Algorithms

Sorting algorithms are fundamental procedures used in computer science to arrange data in a particular order. Among them, Bubble Sort is known as one of the simplest yet illustrative algorithms. It arranges a list by repeatedly stepping through it, comparing adjacent elements, and swapping them if they are in the wrong order. The traditional version of Bubble Sort continues this process until the whole list is sorted, taking multiple passes through the list.

• During each pass, the largest unsorted element "bubbles up" to its correct position.
• The algorithm iterates through the list multiple times until no swaps are needed.
• While easy to understand and implement, Bubble Sort is not the most efficient sorting algorithm available.

To enhance Bubble Sort, a modification is often introduced with an 'isSorted' flag to track whether a pass results in any swaps, thereby potentially reducing unnecessary iterations.

Algorithm Complexity

Algorithm complexity refers to the amount of computational resources used by an algorithm as a function of the input size. In sorting algorithms, we focus on time complexity which assesses how the running time of an algorithm changes with the size of the input.
Bubble Sort is typically described with a time complexity of \(O(n^2)\) in its average and worst cases because each pass through the list involves comparing each element with its neighbor, and for a list of length \(n\), this results in about \(n(n-1)/2\) operations.

• Best Case: If the list is already sorted, the modified Bubble Sort algorithm can stop early, resulting in a time complexity of \(O(n)\).
• Worst Case: Requires performing the full set of comparisons and swaps, leading to \(O(n^2)\) complexity.

Understanding algorithm complexity is crucial for selecting the right sorting technique depending on the specific requirements of the task, such as time constraints and input size.

Programming Problem Solving

Programming problem-solving involves designing algorithms to tackle challenges efficiently and effectively. When working with sorting problems, it is crucial to choose the right algorithm considering both ease of implementation and performance efficiency.
Think of solving a problem like sorting as:

• Understanding the problem requirements and constraints.
• Choosing an algorithm that balances simplicity and efficiency.
• Implementing and optimizing the code for performance gains, like modifying Bubble Sort with an early termination mechanism when the 'isSorted' flag remains true.
• Testing the algorithm with various inputs to ensure it behaves as expected, even in edge cases.

Good problem-solving practices in programming combine theoretical understanding with practical implementation strategies to solve challenges effectively.

Best Case Scenario in Sorting

In sorting algorithms, the best-case scenario is when the input is in its most favorable state, reducing the number of operations needed. For the modified Bubble Sort, the best-case scenario occurs when the list is already sorted.

• The addition of the 'isSorted' flag means if no swaps are needed on the first pass, the algorithm stops early, identifying a sorted list without unnecessary further checks.
• This optimization reduces comparisons, showcasing Bubble Sort’s adaptability to certain input configurations.
• Such efficiency improvements are pivotal in real-world applications where input characteristics vary.

Understanding the best-case scenario aids developers in optimizing algorithms to improve performance on sorted or nearly-sorted data sets, saving both execution time and computational resources.